\usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.5015 rows
The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.
| Identifier | Page | Conf. | LaTeX source | Rendered | Image |
|---|---|---|---|---|---|
| johnston-linear-matrix-algebra_FO0001 | 7 | 1.000 | \cos (\pi / 6)=\sqrt{3} / 2 | | |
| johnston-linear-matrix-algebra_FO0002 | 9 | 1.000 | \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0003 | 9 | 1.000 | \mathbb{C}^{n} | | |
| johnston-linear-matrix-algebra_FO0004 | 9 | 1.000 | \mathbb{F}^{n} | | |
| johnston-linear-matrix-algebra_FO0005 | 9 | 1.000 | \mathbb{F} | | |
| johnston-linear-matrix-algebra_FO0006 | 16 | 1.000 | a \in S | | |
| johnston-linear-matrix-algebra_FO0007 | 16 | 1.000 | a | | |
| johnston-linear-matrix-algebra_FO0008 | 16 | 1.000 | S | | |
| johnston-linear-matrix-algebra_FO0009 | 16 | 1.000 | \mathbf{v} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0010 | 16 | 1.000 | \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0011 | 16 | 1.000 | n | | |
| johnston-linear-matrix-algebra_FO0012 | 16 | 1.000 | 4 x-3=7 | | |
| johnston-linear-matrix-algebra_FO0013 | 16 | 1.000 | f(x)=3 x+8 | | |
| johnston-linear-matrix-algebra_FO0014 | 16 | 1.000 | x | | |
| johnston-linear-matrix-algebra_FO0015 | 16 | 1.000 | 3 x+2 y=1 | | |
| johnston-linear-matrix-algebra_FO0016 | 16 | 0.679 | \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0017 | 16 | 0.644 | \mathbf{v}=(2,3) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0018 | 16 | 0.611 | \mathbf{w}=(1,3,2) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0019 | 16 | 0.611 | 4 \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0020 | 16 | 0.968 | y | | |
| johnston-linear-matrix-algebra_FO0021 | 16 | 1.000 | z | | |
| johnston-linear-matrix-algebra_FO0022 | 17 | 1.000 | \vec{v} | | |
| johnston-linear-matrix-algebra_FO0023 | 17 | 1.000 | \overrightarrow{A B} | | |
| johnston-linear-matrix-algebra_FO0024 | 17 | 1.000 | A | | |
| johnston-linear-matrix-algebra_FO0025 | 17 | 1.000 | B | | |
| johnston-linear-matrix-algebra_FO0026 | 17 | 0.968 | (-1,1) | | |
| johnston-linear-matrix-algebra_FO0027 | 17 | 0.968 | (2,2) | | |
| johnston-linear-matrix-algebra_FO0028 | 17 | 0.968 | (2,2)-(-1,1)=(3,1) | | |
| johnston-linear-matrix-algebra_FO0029 | 17 | 0.395 | (4,1)-(1,0)=(3,1) | | |
| johnston-linear-matrix-algebra_FO0031 | 17 | 1.000 | \mathbb{R}^{7} | | |
| johnston-linear-matrix-algebra_FO0032 | 17 | 1.000 | \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0033 | 18 | 1.000 | \mathbf{v}+\mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0034 | 18 | 1.000 | \mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right) \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0035 | 18 | 1.000 | \mathbf{w}=\left(w_{1}, w_{2}, \ldots, w_{n}\right) \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0036 | 18 | 1.000 | \mathbf{v}+ | | |
| johnston-linear-matrix-algebra_FO0037 | 18 | 1.000 | \mathbf{v}, \mathbf{w}, \mathbf{x} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0038 | 18 | 1.000 | \mathbf{v}+\mathbf{w}=\mathbf{w}+\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0039 | 18 | 1.000 | (\mathbf{v}+\mathbf{w})+\mathbf{x}=\mathbf{v}+(\mathbf{w}+\mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0040 | 18 | 1.000 | x+y=y+x | | |
| johnston-linear-matrix-algebra_FO0041 | 18 | 1.000 | x, y \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0043 | 19 | 1.000 | \mathbf{v}+\mathbf{w}+\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0044 | 19 | 1.000 | (\mathbf{v}+\mathbf{w})+\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0045 | 19 | 1.000 | \mathbf{v}+(\mathbf{w}+\mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0046 | 19 | 1.000 | \mathbf{v} \times \mathbf{w} \neq \mathbf{w} \times \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0047 | 19 | 1.000 | (2,5,-1)+(1,-1,2) | | |
| johnston-linear-matrix-algebra_FO0048 | 19 | 1.000 | (1,2)+(3,1)+(2,-1) | | |
| johnston-linear-matrix-algebra_FO0049 | 19 | 0.672 | 0,0,0 | | |
| johnston-linear-matrix-algebra_FO0050 | 19 | 0.672 | 1,1,1 | | |
| johnston-linear-matrix-algebra_FO0051 | 19 | 1.000 | (2,5,-1)+(1,-1,2)=(2+1,5-1,-1+2)=(3,4,1) | | |
| johnston-linear-matrix-algebra_FO0052 | 19 | 1.000 | (1,2)+(3,1)+(2,-1)=(1+3+2,2+1-1)=(6,2) | | |
| johnston-linear-matrix-algebra_FO0053 | 20 | 1.000 | c \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0054 | 20 | 1.000 | c \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0055 | 20 | 0.531 | c | | |
| johnston-linear-matrix-algebra_FO0056 | 20 | 0.967 | |c|>1 | | |
| johnston-linear-matrix-algebra_FO0057 | 20 | 0.967 | |c|<1 | | |
| johnston-linear-matrix-algebra_FO0058 | 20 | 0.967 | c<0 | | |
| johnston-linear-matrix-algebra_FO0059 | 20 | 1.000 | |c| \neq 1 | | |
| johnston-linear-matrix-algebra_FO0060 | 20 | 0.772 | c=0 | | |
| johnston-linear-matrix-algebra_FO0061 | 20 | 1.000 | \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0062 | 20 | 0.957 | c=-1 | | |
| johnston-linear-matrix-algebra_FO0063 | 20 | 0.637 | \mathrm{via} \mathbf{v}-\mathbf{w} \stackrel{\text { def }}{=} \mathbf{v}+(-\mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0064 | 20 | 1.000 | \mathbf{v}-\mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0065 | 21 | 1.000 | \mathbf{v}-\mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0066 | 21 | 1.000 | \mathbf{v}+\mathbf{0}=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0067 | 21 | 1.000 | \mathbf{v}, \mathbf{w} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0068 | 21 | 1.000 | c, d \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0069 | 21 | 1.000 | c(\mathbf{v}+\mathbf{w})=c \mathbf{v}+c \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0070 | 21 | 1.000 | (c+d) \mathbf{v}=c \mathbf{v}+d \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0071 | 21 | 1.000 | c(d \mathbf{v})=(c d) \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0072 | 21 | 1.000 | 3 \mathbf{v}-2 \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0073 | 21 | 1.000 | \mathbf{v}=(2,1,-1) | | |
| johnston-linear-matrix-algebra_FO0074 | 21 | 1.000 | \mathbf{w}=(-1,0,3) | | |
| johnston-linear-matrix-algebra_FO0075 | 21 | 1.000 | (0,0) | | |
| johnston-linear-matrix-algebra_FO0076 | 22 | 1.000 | \mathbf{x}=(1,1,1)) | | |
| johnston-linear-matrix-algebra_FO0077 | 22 | 0.968 | \mathbf{4 x}=(4,4,4)) | | |
| johnston-linear-matrix-algebra_FO0078 | 22 | 1.000 | 3 \mathbf{v}-2 \mathbf{w}=(6,3,-3)-(-2,0,6)=(8,3,-9) | | |
| johnston-linear-matrix-algebra_FO0079 | 22 | 0.980 | \mathbf{v}, \mathbf{w}, \mathbf{x},-\mathbf{v},-\mathbf{w},-\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0080 | 22 | 1.000 | \mathbf{v}+\mathbf{w}+\mathbf{x}-\mathbf{v}-\mathbf{w}-\mathbf{x}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0081 | 22 | 0.972 | \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0082 | 22 | 1.000 | \mathbf{x}-(3,2,1)=(1,2,3)-3 \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0083 | 22 | 1.000 | \mathbf{x}+2(\mathbf{v}+\mathbf{w})=-\mathbf{v}-3(\mathbf{x}-\mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0084 | 22 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO0085 | 23 | 1.000 | \mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO0086 | 23 | 0.574 | \mathbf{e}_{3} \in \mathbb{R}^{7} | | |
| johnston-linear-matrix-algebra_FO0087 | 23 | 0.999 | v_{1}, v_{2}, \ldots, v_{n} | | |
| johnston-linear-matrix-algebra_FO0088 | 23 | 1.000 | \mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n} | | |
| johnston-linear-matrix-algebra_FO0089 | 23 | 0.997 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0090 | 23 | 0.999 | c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0091 | 23 | 0.647 | (-1,0,1) | | |
| johnston-linear-matrix-algebra_FO0092 | 23 | 0.647 | (1,2,3)=2(1,1,1)+(-1,0,1) | | |
| johnston-linear-matrix-algebra_FO0093 | 23 | 0.647 | (1,2,3) | | |
| johnston-linear-matrix-algebra_FO0094 | 23 | 1.000 | c_{1}(1,1,0)+c_{2}(2,1,0) | | |
| johnston-linear-matrix-algebra_FO0095 | 23 | 1.000 | j=1,2, \ldots, n | | |
| johnston-linear-matrix-algebra_FO0096 | 23 | 1.000 | \mathbf{e}_{j} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0097 | 23 | 1.000 | \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0098 | 23 | 1.000 | \mathbf{e}_{1}=(1,0) | | |
| johnston-linear-matrix-algebra_FO0099 | 23 | 1.000 | \mathbf{e}_{2}=(0,1) | | |
| johnston-linear-matrix-algebra_FO0100 | 23 | 1.000 | \mathbf{e}_{1}=(1,0,0), \mathbf{e}_{2}=(0,1,0) | | |
| johnston-linear-matrix-algebra_FO0101 | 23 | 1.000 | \mathbf{e}_{3}=(0,0,1) | | |
| johnston-linear-matrix-algebra_FO0103 | 23 | 1.000 | \mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right) | | |
| johnston-linear-matrix-algebra_FO0104 | 24 | 1.000 | 3 \mathbf{e}_{1}-2 \mathbf{e}_{2}+\mathbf{e}_{3} \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0105 | 24 | 0.973 | \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}, \mathbf{e}_{4} \in \mathbb{R}^{4} | | |
| johnston-linear-matrix-algebra_FO0106 | 24 | 1.000 | 3 \mathbf{e}_{1}-2 \mathbf{e}_{2}+\mathbf{e}_{3}=3(1,0,0)-2(0,1,0)+(0,0,1)=(3,-2,1) | | |
| johnston-linear-matrix-algebra_FO0107 | 24 | 1.000 | \mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO0108 | 24 | 1.000 | \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO0109 | 24 | 1.000 | (3,5,-2,-1)=3 \mathbf{e}_{1}+5 \mathbf{e}_{2}-2 \mathbf{e}_{3}-\mathbf{e}_{4} | | |
| johnston-linear-matrix-algebra_FO0110 | 24 | 1.000 | \mathbf{v}=(3,2) | | |
| johnston-linear-matrix-algebra_FO0111 | 24 | 1.000 | \mathbf{w}=(-0.5,3) | | |
| johnston-linear-matrix-algebra_FO0112 | 24 | 0.998 | \mathbf{x}=(1,-3) | | |
| johnston-linear-matrix-algebra_FO0113 | 24 | 1.000 | \mathbf{y}=(-2,-1) | | |
| johnston-linear-matrix-algebra_FO0114 | 24 | 1.000 | \mathbf{v}=(0,0,2) | | |
| johnston-linear-matrix-algebra_FO0115 | 24 | 1.000 | \mathbf{w}=(-1,2,1) | | |
| johnston-linear-matrix-algebra_FO0116 | 24 | 0.999 | \mathbf{x}=(1,2,0) | | |
| johnston-linear-matrix-algebra_FO0117 | 24 | 1.000 | \mathbf{y}=(3,2,-1) | | |
| johnston-linear-matrix-algebra_FO0118 | 24 | 1.000 | \mathbf{v}, \mathbf{w}, \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0119 | 24 | 1.000 | \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO0120 | 24 | 1.000 | \mathbf{v}+\mathbf{w}+\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO0121 | 24 | 0.967 | \mathbf{y}-2 \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0122 | 24 | 1.000 | \mathbf{v}+2 \mathbf{w}+2 \mathbf{x}+2 \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO0123 | 24 | 1.000 | \mathbf{v}+\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO0124 | 24 | 1.000 | 4 \mathbf{w}+3 \mathbf{w}-(2 \mathbf{w}+6 \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0125 | 24 | 0.979 | 4 \mathbf{x}-2 \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0126 | 24 | 0.992 | 2 \mathbf{x}-\mathbf{w}-\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO0127 | 24 | 1.000 | \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3} \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0128 | 24 | 1.000 | \mathbf{v}=(1,4) | | |
| johnston-linear-matrix-algebra_FO0129 | 24 | 1.000 | \mathbf{w}=(-2,1) | | |
| johnston-linear-matrix-algebra_FO0130 | 24 | 1.000 | \mathbf{x}=(3,-2) | | |
| johnston-linear-matrix-algebra_FO0131 | 24 | 1.000 | \mathbf{y}=(1,4) | | |
| johnston-linear-matrix-algebra_FO0132 | 24 | 0.630 | (1,2)-\mathbf{x}=(3,4)-2 \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0133 | 24 | 1.000 | 3((1,-1)+\mathbf{x})=2 \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0134 | 24 | 1.000 | 2(\mathbf{x}+2(\mathbf{x}+2 \mathbf{x}))=3(\mathbf{x}+3(\mathbf{x}+3 \mathbf{x})) | | |
| johnston-linear-matrix-algebra_FO0135 | 24 | 1.000 | -2(\mathbf{x}-(1,-2))=\mathbf{x}+2(\mathbf{x}+(1,1)) | | |
| johnston-linear-matrix-algebra_FO0136 | 25 | 1.000 | \mathbf{v}-\mathbf{x}=\mathbf{w}+\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0137 | 25 | 1.000 | 2 \mathbf{v}-3 \mathbf{x}=4 \mathbf{x}-5 \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0138 | 25 | 1.000 | 4(\mathbf{x}+\mathbf{v})-\mathbf{x}=2(\mathbf{w}+\mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0139 | 25 | 1.000 | 2(\mathbf{x}+2(\mathbf{x}+2 \mathbf{x}))=2(\mathbf{v}+2 \mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0140 | 25 | 0.856 | c(1,2)= | | |
| johnston-linear-matrix-algebra_FO0141 | 25 | 1.000 | n \geq 3 | | |
| johnston-linear-matrix-algebra_FO0142 | 25 | 0.844 | n=6 | | |
| johnston-linear-matrix-algebra_FO0143 | 25 | 0.985 | \mathbf{v} \cdot \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0144 | 25 | 1.000 | \mathbf{v} \cdot(\mathbf{w} \cdot \mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0145 | 25 | 0.995 | \mathbf{w} \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0146 | 25 | 1.000 | \mathbf{v} /(\mathbf{w} \cdot \mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0147 | 25 | 1.000 | (1,2,3) \cdot(4,-3,2) | | |
| johnston-linear-matrix-algebra_FO0148 | 25 | 0.945 | (3,6,2) \cdot(-1,5,2,1) | | |
| johnston-linear-matrix-algebra_FO0149 | 26 | 1.000 | j | | |
| johnston-linear-matrix-algebra_FO0150 | 26 | 1.000 | \left(v_{1}, v_{2}, \ldots, v_{n}\right) \cdot \mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO0151 | 26 | 1.000 | 1 \leq j \leq n | | |
| johnston-linear-matrix-algebra_FO0152 | 26 | 1.000 | (1,2,3) \cdot(4,-3,2)=1 \cdot 4+2 \cdot(-3)+3 \cdot 2=4-6+6=4 | | |
| johnston-linear-matrix-algebra_FO0153 | 26 | 0.997 | \left(v_{1}, v_{2}, \ldots, v_{n}\right) | | |
| johnston-linear-matrix-algebra_FO0154 | 26 | 0.996 | \mathbf{v}=\mathbf{w}=(1,0) | | |
| johnston-linear-matrix-algebra_FO0155 | 26 | 0.755 | \mathbf{v} \cdot \mathbf{w}=1 | | |
| johnston-linear-matrix-algebra_FO0156 | 26 | 0.982 | \mathbf{w}=(0,1) | | |
| johnston-linear-matrix-algebra_FO0157 | 26 | 0.982 | \mathbf{w}=(0,-1) | | |
| johnston-linear-matrix-algebra_FO0158 | 26 | 0.963 | \mathbf{w}=(-1,0) | | |
| johnston-linear-matrix-algebra_FO0159 | 26 | 1.000 | \theta | | |
| johnston-linear-matrix-algebra_FO0160 | 26 | 1.000 | \mathbf{w}=(\cos (\theta), \sin (\theta)) | | |
| johnston-linear-matrix-algebra_FO0161 | 26 | 1.000 | \mathbf{v} \cdot \mathbf{w}=1 \cos (\theta)+0 \sin (\theta)=\cos (\theta) | | |
| johnston-linear-matrix-algebra_FO0163 | 26 | 1.000 | \cos (\theta) | | |
| johnston-linear-matrix-algebra_FO0164 | 26 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\mathbf{w} \cdot \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0165 | 26 | 1.000 | \mathbf{v} \cdot(\mathbf{w}+\mathbf{x})=\mathbf{v} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0166 | 26 | 1.000 | \mathbf{v} \cdot(c \mathbf{w})=c(\mathbf{v} \cdot \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0167 | 27 | 1.000 | (x+2)\left(x^{2}+3 x\right) | | |
| johnston-linear-matrix-algebra_FO0168 | 27 | 1.000 | b | | |
| johnston-linear-matrix-algebra_FO0169 | 27 | 1.000 | c^{2}=a^{2}+b^{2} | | |
| johnston-linear-matrix-algebra_FO0170 | 27 | 1.000 | \left.c=\sqrt{a^{2}+b^{2}}\right) | | |
| johnston-linear-matrix-algebra_FO0171 | 27 | 0.992 | \|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0172 | 27 | 0.992 | \mathbf{v}=\left(v_{1}, v_{2}\right) \in | | |
| johnston-linear-matrix-algebra_FO0173 | 27 | 1.000 | \mathbf{v}=\left(v_{1}, 0\right)+\left(0, v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0174 | 27 | 1.000 | \left(v_{1}, 0\right) | | |
| johnston-linear-matrix-algebra_FO0175 | 27 | 1.000 | \left(0, v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0176 | 27 | 1.000 | \left|v_{1}\right| | | |
| johnston-linear-matrix-algebra_FO0177 | 27 | 1.000 | \left|v_{2}\right| | | |
| johnston-linear-matrix-algebra_FO0178 | 27 | 1.000 | \mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0179 | 27 | 1.000 | \mathbf{v}=\left(v_{1}, v_{2}, 0\right)+\left(0,0, v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO0180 | 27 | 0.995 | \left(v_{1}, v_{2}, 0\right) | | |
| johnston-linear-matrix-algebra_FO0181 | 27 | 0.995 | \left(0,0, v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO0182 | 27 | 1.000 | \sqrt{v_{1}^{2}+v_{2}^{2}} | | |
| johnston-linear-matrix-algebra_FO0183 | 28 | 1.000 | \left|v_{3}\right| | | |
| johnston-linear-matrix-algebra_FO0184 | 28 | 1.000 | \mathbf{v} \cdot \mathbf{v}=v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2} | | |
| johnston-linear-matrix-algebra_FO0185 | 28 | 0.963 | (\cos (\theta), \sin (\theta)) | | |
| johnston-linear-matrix-algebra_FO0186 | 28 | 1.000 | \|(2,-5,4,6)\|=\sqrt{2^{2}+(-5)^{2}+4^{2}+6^{2}}=\sqrt{81}=9 | | |
| johnston-linear-matrix-algebra_FO0187 | 28 | 1.000 | \|(\cos (\theta), \sin (\theta))\|=\sqrt{\cos ^{2}(\theta)+\sin ^{2}(\theta)}=\sqrt{1}=1 | | |
| johnston-linear-matrix-algebra_FO0188 | 29 | 1.000 | \sqrt{c^{2}}=|c| | | |
| johnston-linear-matrix-algebra_FO0189 | 29 | 1.000 | v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2} | | |
| johnston-linear-matrix-algebra_FO0190 | 29 | 1.000 | \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0191 | 29 | 1.000 | \mathbf{v}=\|\mathbf{v}\| \mathbf{u} | | |
| johnston-linear-matrix-algebra_FO0192 | 29 | 1.000 | \mathbf{u} | | |
| johnston-linear-matrix-algebra_FO0193 | 29 | 1.000 | \mathbf{v}=(1,1,1) | | |
| johnston-linear-matrix-algebra_FO0194 | 29 | 1.000 | \|\mathbf{v}\|=\sqrt{1^{2}+1^{2}+1^{2}}=\sqrt{3} | | |
| johnston-linear-matrix-algebra_FO0195 | 29 | 0.998 | \|c \mathbf{v}\|=|c|\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0196 | 29 | 1.000 | \|\mathbf{v}\| \geq 0 | | |
| johnston-linear-matrix-algebra_FO0197 | 29 | 1.000 | \|\mathbf{0}\|=0 | | |
| johnston-linear-matrix-algebra_FO0198 | 29 | 1.000 | \|\mathbf{v}\|=0 | | |
| johnston-linear-matrix-algebra_FO0199 | 29 | 1.000 | v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}=0 | | |
| johnston-linear-matrix-algebra_FO0200 | 29 | 1.000 | v_{j}^{2} \geq 0 | | |
| johnston-linear-matrix-algebra_FO0201 | 29 | 1.000 | v_{j}=0 | | |
| johnston-linear-matrix-algebra_FO0202 | 29 | 1.000 | v_{1}=v_{2}=\cdots=v_{n}=0 | | |
| johnston-linear-matrix-algebra_FO0203 | 30 | 1.000 | \mathbf{u}=\mathbf{v} /\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0204 | 30 | 1.000 | \mathbf{v}=(1,0) | | |
| johnston-linear-matrix-algebra_FO0205 | 30 | 1.000 | |\mathbf{v} \cdot \mathbf{w}| \leq\|\mathbf{v}\|\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0206 | 30 | 0.994 | \|c \mathbf{v}+d \mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO0207 | 30 | 1.000 | d | | |
| johnston-linear-matrix-algebra_FO0208 | 30 | 1.000 | \mathbf{w}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0209 | 30 | 1.000 | 0 \leq 0 | | |
| johnston-linear-matrix-algebra_FO0210 | 30 | 1.000 | c=\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0211 | 30 | 1.000 | d=-(\mathbf{v} \cdot \mathbf{w}) /\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0212 | 30 | 1.000 | \|\mathbf{v}+\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0213 | 30 | 1.000 | \|\mathbf{v}\|+\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0214 | 30 | 1.000 | \|\mathbf{v}+\mathbf{w}\| \leq\|\mathbf{v}\|+\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0215 | 31 | 1.000 | a, b | | |
| johnston-linear-matrix-algebra_FO0216 | 31 | 1.000 | \|\mathbf{v}+\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO0217 | 31 | 0.971 | \mathbf{v}, \mathbf{w} \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0218 | 31 | 1.000 | \mathbf{v}, \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0219 | 31 | 1.000 | \|\mathbf{v}\|,\|\mathbf{w}\|,\|\mathbf{v}-\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0220 | 32 | 1.000 | 0 \leq \theta \leq \pi | | |
| johnston-linear-matrix-algebra_FO0221 | 32 | 1.000 | \arccos (x)=\theta | | |
| johnston-linear-matrix-algebra_FO0222 | 32 | 1.000 | \cos (\theta)=x | | |
| johnston-linear-matrix-algebra_FO0223 | 32 | 1.000 | \cos ^{-1} | | |
| johnston-linear-matrix-algebra_FO0224 | 32 | 1.000 | \|\mathbf{v}\|,\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0226 | 32 | 1.000 | \|\mathbf{v}-\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO0227 | 32 | 0.651 | \mathbf{v}, \mathbf{w} \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0228 | 32 | 1.000 | (\mathbf{v} \cdot \mathbf{w}) /(\|\mathbf{v}\|\|\mathbf{w}\|) | | |
| johnston-linear-matrix-algebra_FO0229 | 32 | 1.000 | \mathbf{v}=(1,2) | | |
| johnston-linear-matrix-algebra_FO0230 | 32 | 1.000 | \mathbf{w}=(3,4) | | |
| johnston-linear-matrix-algebra_FO0231 | 32 | 1.000 | \mathbf{v}=(1,2,-1,-2) | | |
| johnston-linear-matrix-algebra_FO0232 | 32 | 1.000 | \mathbf{w}=(1,-1,1,-1) | | |
| johnston-linear-matrix-algebra_FO0233 | 33 | 1.000 | \arccos (11 /(5 \sqrt{5})) | | |
| johnston-linear-matrix-algebra_FO0234 | 33 | 1.000 | \arccos (x) | | |
| johnston-linear-matrix-algebra_FO0235 | 33 | 1.000 | \arccos (\sqrt{0} / 2)=\pi / 2 | | |
| johnston-linear-matrix-algebra_FO0236 | 33 | 1.000 | \arccos (\sqrt{1} / 2)=\pi / 3 | | |
| johnston-linear-matrix-algebra_FO0237 | 33 | 1.000 | \arccos (\sqrt{2} / 2)=\pi / 4 | | |
| johnston-linear-matrix-algebra_FO0238 | 33 | 1.000 | \arccos (\sqrt{3} / 2)=\pi / 6 | | |
| johnston-linear-matrix-algebra_FO0239 | 33 | 1.000 | \arccos (\sqrt{4} / 2)=0 | | |
| johnston-linear-matrix-algebra_FO0240 | 33 | 1.000 | \mathbf{v} \cdot \mathbf{w}=3+8=11,\|\mathbf{v}\|=\sqrt{5} | | |
| johnston-linear-matrix-algebra_FO0241 | 33 | 1.000 | \|\mathbf{w}\|=5 | | |
| johnston-linear-matrix-algebra_FO0242 | 33 | 1.000 | \mathbf{v} \cdot \mathbf{w}=1-2-1+2=0 | | |
| johnston-linear-matrix-algebra_FO0243 | 33 | 1.000 | \|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0244 | 33 | 0.999 | \mathbf{v} \cdot \mathbf{w}=0 | | |
| johnston-linear-matrix-algebra_FO0245 | 33 | 1.000 | \theta=\pi / 2 | | |
| johnston-linear-matrix-algebra_FO0246 | 33 | 0.503 | (0,0,0) | | |
| johnston-linear-matrix-algebra_FO0247 | 33 | 0.503 | (1,1,1) | | |
| johnston-linear-matrix-algebra_FO0248 | 33 | 1.000 | \mathbf{v}=(1,0,1)-(1,1,0)=(0,-1,1) | | |
| johnston-linear-matrix-algebra_FO0249 | 33 | 1.000 | \mathbf{w}=(0,1,1)-(1,1,0)=(-1,0,1) | | |
| johnston-linear-matrix-algebra_FO0250 | 33 | 1.000 | \mathbf{v} \cdot \mathbf{w}=0+0+1=1,\|\mathbf{v}\|=\sqrt{2} | | |
| johnston-linear-matrix-algebra_FO0251 | 33 | 1.000 | \|\mathbf{w}\|=\sqrt{2} | | |
| johnston-linear-matrix-algebra_FO0252 | 33 | 0.988 | \pi / 2 | | |
| johnston-linear-matrix-algebra_FO0253 | 33 | 0.953 | \arccos (0)=\pi / 2 | | |
| johnston-linear-matrix-algebra_FO0254 | 33 | 1.000 | \mathbf{v}, \mathbf{w} \in \mathbb{R}^{4} | | |
| johnston-linear-matrix-algebra_FO0255 | 34 | 0.998 | \mathbf{v}=(-2,4), \mathbf{w}=(2,1) | | |
| johnston-linear-matrix-algebra_FO0256 | 34 | 1.000 | \mathbf{v}=(1,2,3), \mathbf{w}=(-3,2,-1) | | |
| johnston-linear-matrix-algebra_FO0257 | 34 | 0.998 | \mathbf{v}=(3,-1,0,1), \mathbf{w}=(0,2,1,3) | | |
| johnston-linear-matrix-algebra_FO0258 | 34 | 1.000 | \mathbf{v}=(\sqrt{2}, \sqrt{3}, \sqrt{5}), \mathbf{w}=(\sqrt{2}, \sqrt{3}, \sqrt{5}) | | |
| johnston-linear-matrix-algebra_FO0259 | 34 | 0.999 | \mathbf{v}=\mathbf{0} \in \mathbb{R}^{9}, \mathbf{w}=(8,1,5,-7,3,9,1,-3,2) | | |
| johnston-linear-matrix-algebra_FO0260 | 34 | 0.999 | \mathbf{v}=(3,4) | | |
| johnston-linear-matrix-algebra_FO0261 | 34 | 1.000 | \mathbf{v}=(2,1,-2) | | |
| johnston-linear-matrix-algebra_FO0262 | 34 | 1.000 | \mathbf{v}=(-2 \sqrt{2},-3, \sqrt{10}, 3) | | |
| johnston-linear-matrix-algebra_FO0263 | 34 | 1.000 | \mathbf{v}=(\cos (\theta), \sin (\theta)) | | |
| johnston-linear-matrix-algebra_FO0264 | 34 | 1.000 | \mathbf{v}=(1, \sqrt{3}), \mathbf{w}=(\sqrt{3}, 1) | | |
| johnston-linear-matrix-algebra_FO0265 | 34 | 1.000 | \mathbf{v}=(0,-2,2), \mathbf{w}=(1,0,1) | | |
| johnston-linear-matrix-algebra_FO0266 | 34 | 1.000 | \mathbf{v}=(1,1,1,1), \mathbf{w}=(-2,0,-2,0) | | |
| johnston-linear-matrix-algebra_FO0267 | 34 | 1.000 | \mathbf{v}=(2,1,-3), \mathbf{w}=(-1,2,3) | | |
| johnston-linear-matrix-algebra_FO0268 | 34 | 0.842 | \mathbf{v}=(\cos (\boldsymbol{\theta}), \sin (\boldsymbol{\theta})), \mathbf{w}=(-\sin (\boldsymbol{\theta}), \cos (\boldsymbol{\theta})) | | |
| johnston-linear-matrix-algebra_FO0269 | 34 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\mathbf{v} \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0270 | 34 | 0.785 | \mathbf{w}=\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0271 | 34 | 1.000 | \mathbf{w} \cdot \mathbf{x}=0 | | |
| johnston-linear-matrix-algebra_FO0272 | 34 | 1.000 | \mathbf{v} \cdot \mathbf{x}=0 | | |
| johnston-linear-matrix-algebra_FO0273 | 34 | 0.760 | \|\mathbf{v}\|+\|\mathbf{w}\| \leq 2 | | |
| johnston-linear-matrix-algebra_FO0274 | 34 | 1.000 | \|\mathbf{v}+\mathbf{w}\| \leq 2 | | |
| johnston-linear-matrix-algebra_FO0275 | 34 | 1.000 | \mathbf{v}, \mathbf{w} \in \mathbb{R}^{5} | | |
| johnston-linear-matrix-algebra_FO0276 | 34 | 1.000 | \|\mathbf{v}\|=2 | | |
| johnston-linear-matrix-algebra_FO0277 | 34 | 1.000 | \|\mathbf{w}\|=4 | | |
| johnston-linear-matrix-algebra_FO0278 | 34 | 1.000 | \|\mathbf{v}-\mathbf{w}\|=1 | | |
| johnston-linear-matrix-algebra_FO0279 | 34 | 0.995 | \|\mathbf{v}\|=1 | | |
| johnston-linear-matrix-algebra_FO0280 | 34 | 1.000 | \|\mathbf{w}\|=2 | | |
| johnston-linear-matrix-algebra_FO0281 | 34 | 1.000 | \mathbf{v} \cdot \mathbf{w}=-1 | | |
| johnston-linear-matrix-algebra_FO0282 | 34 | 1.000 | |\mathbf{v} \cdot \mathbf{w}| \leq 1 | | |
| johnston-linear-matrix-algebra_FO0283 | 34 | 0.598 | \|\mathbf{v}\| \leq 1 | | |
| johnston-linear-matrix-algebra_FO0284 | 34 | 1.000 | \|\mathbf{w}\| \leq 1 | | |
| johnston-linear-matrix-algebra_FO0285 | 34 | 0.965 | \mathbf{v}=(3, \sqrt{3}) | | |
| johnston-linear-matrix-algebra_FO0286 | 34 | 0.965 | \mathbf{w} \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0287 | 34 | 1.000 | \theta=\pi / 3 | | |
| johnston-linear-matrix-algebra_FO0288 | 34 | 0.997 | \mathbf{v} \cdot(\mathbf{w}-\mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0289 | 34 | 1.000 | (\mathbf{v} \cdot \mathbf{w}) \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0290 | 34 | 1.000 | \mathbf{v}+(\mathbf{w} \cdot \mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0291 | 34 | 1.000 | \mathbf{v} /\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0292 | 34 | 0.647 | *(\mathrm{e}) \mathrm{v}^{2} | | |
| johnston-linear-matrix-algebra_FO0293 | 34 | 1.000 | (\mathbf{v}+\mathbf{w}) / \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0294 | 34 | 1.000 | \left\|\mathbf{e}_{j}\right\| | | |
| johnston-linear-matrix-algebra_FO0295 | 34 | 1.000 | 1 \leq i, j \leq n | | |
| johnston-linear-matrix-algebra_FO0296 | 34 | 1.000 | \mathbf{e}_{i} \cdot \mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO0297 | 34 | 1.000 | i=j | | |
| johnston-linear-matrix-algebra_FO0298 | 34 | 1.000 | i \neq j | | |
| johnston-linear-matrix-algebra_FO0299 | 34 | 1.000 | \mathbf{v}=(1,2) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0300 | 34 | 1.000 | \mathbf{v}=(1,2,3) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0301 | 35 | 1.000 | \mathbf{v}, \mathbf{w} \in \mathbb{C}^{n} | | |
| johnston-linear-matrix-algebra_FO0302 | 35 | 1.000 | \overline{a+i b}=a-i b | | |
| johnston-linear-matrix-algebra_FO0303 | 35 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\overline{\mathbf{w} \cdot \mathbf{v}} | | |
| johnston-linear-matrix-algebra_FO0304 | 35 | 0.988 | c \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO0305 | 35 | 0.988 | (c \mathbf{v}) \cdot \mathbf{w}=\bar{c}(\mathbf{v} \cdot \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0306 | 35 | 1.000 | \mathbf{x} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0307 | 35 | 1.000 | \mathbf{x} \cdot \mathbf{y}=0 | | |
| johnston-linear-matrix-algebra_FO0308 | 35 | 1.000 | \mathbf{y} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0309 | 35 | 1.000 | \mathbf{x}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0310 | 35 | 1.000 | |\mathbf{v} \cdot \mathbf{w}|=\|\mathbf{v}\|\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0311 | 35 | 1.000 | \mathbf{v}=c \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0312 | 35 | 1.000 | \|\mathbf{v}+\mathbf{w}\|=\|\mathbf{v}\|+\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0313 | 35 | 1.000 | 0 \leq c \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0314 | 35 | 1.000 | \|\mathbf{v}+\mathbf{w}\|^{2}=\|\mathbf{v}\|^{2}+\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO0315 | 35 | 1.000 | \|\mathbf{v}+\mathbf{w}\|^{2}+\|\mathbf{v}-\mathbf{w}\|^{2}=2\|\mathbf{v}\|^{2}+2\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO0316 | 35 | 1.000 | \mathbf{v} \cdot \mathbf{w}= | | |
| johnston-linear-matrix-algebra_FO0317 | 35 | 1.000 | \frac{1}{4}\left(\|\mathbf{v}+\mathbf{w}\|^{2}-\|\mathbf{v}-\mathbf{w}\|^{2}\right) | | |
| johnston-linear-matrix-algebra_FO0318 | 35 | 0.816 | \|\mathbf{v}-\mathbf{w}\| \geq | | |
| johnston-linear-matrix-algebra_FO0319 | 35 | 1.000 | \|\mathbf{v}\|-\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0320 | 35 | 1.000 | d=\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0321 | 35 | 1.000 | d=-\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0322 | 35 | 1.000 | f(x)=\|\mathbf{v}-x \mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO0323 | 35 | 1.000 | a x^{2}+b x+c | | |
| johnston-linear-matrix-algebra_FO0324 | 35 | 1.000 | b^{2}-4 a c | | |
| johnston-linear-matrix-algebra_FO0325 | 35 | 0.777 | \leq 0 | | |
| johnston-linear-matrix-algebra_FO0326 | 35 | 1.000 | f | | |
| johnston-linear-matrix-algebra_FO0327 | 35 | 1.000 | |x+y| \leq|x|+|y| | | |
| johnston-linear-matrix-algebra_FO0328 | 35 | 1.000 | x_{1}, \ldots, x_{n} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0329 | 35 | 1.000 | \left|x_{1}+\cdots+x_{n}\right| \leq | | |
| johnston-linear-matrix-algebra_FO0330 | 35 | 1.000 | \left|x_{1}\right|+\cdots+\left|x_{n}\right| | | |
| johnston-linear-matrix-algebra_FO0331 | 35 | 1.000 | x y \leq \frac{1}{2}\left(x^{2}+y^{2}\right) | | |
| johnston-linear-matrix-algebra_FO0332 | 35 | 1.000 | |\mathbf{v} \cdot \mathbf{w}| /(\|\mathbf{v}\|\|\mathbf{w}\|) \leq 1 | | |
| johnston-linear-matrix-algebra_FO0333 | 35 | 1.000 | \mathbf{v} \cdot \mathbf{w}=v_{1} w_{1}+\cdots+v_{n} w_{n} | | |
| johnston-linear-matrix-algebra_FO0334 | 36 | 1.000 | a_{i, j} | | |
| johnston-linear-matrix-algebra_FO0335 | 36 | 1.000 | i | | |
| johnston-linear-matrix-algebra_FO0336 | 36 | 0.623 | 3 \times 4 | | |
| johnston-linear-matrix-algebra_FO0337 | 36 | 1.000 | A, B, C, \ldots | | |
| johnston-linear-matrix-algebra_FO0338 | 36 | 1.000 | a_{1,2}=3 | | |
| johnston-linear-matrix-algebra_FO0339 | 36 | 0.998 | a_{2,1}=2 | | |
| johnston-linear-matrix-algebra_FO0340 | 36 | 0.998 | b_{2,3}=1 | | |
| johnston-linear-matrix-algebra_FO0341 | 36 | 0.999 | (i, j) | | |
| johnston-linear-matrix-algebra_FO0342 | 36 | 1.000 | [A]_{i, j} | | |
| johnston-linear-matrix-algebra_FO0343 | 36 | 0.633 | b_{2,1}=[B]_{2,1}=0 | | |
| johnston-linear-matrix-algebra_FO0344 | 36 | 1.000 | m | | |
| johnston-linear-matrix-algebra_FO0345 | 36 | 1.000 | \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO0346 | 36 | 1.000 | \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO0347 | 36 | 1.000 | m=n | | |
| johnston-linear-matrix-algebra_FO0348 | 36 | 1.000 | A \in \mathcal{M}_{2} | | |
| johnston-linear-matrix-algebra_FO0349 | 36 | 1.000 | B \in \mathcal{M}_{2,3} | | |
| johnston-linear-matrix-algebra_FO0350 | 36 | 1.000 | A, B \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO0351 | 36 | 0.985 | A+B | | |
| johnston-linear-matrix-algebra_FO0352 | 36 | 0.985 | c A | | |
| johnston-linear-matrix-algebra_FO0353 | 36 | 0.985 | m \times n | | |
| johnston-linear-matrix-algebra_FO0354 | 36 | 1.000 | 1 \leq i \leq m | | |
| johnston-linear-matrix-algebra_FO0355 | 36 | 1.000 | O | | |
| johnston-linear-matrix-algebra_FO0356 | 36 | 1.000 | O_{m, n} | | |
| johnston-linear-matrix-algebra_FO0357 | 36 | 1.000 | O_{n} | | |
| johnston-linear-matrix-algebra_FO0358 | 36 | 1.000 | n \times n | | |
| johnston-linear-matrix-algebra_FO0359 | 36 | 0.971 | (A-B=A+(-1) B) | | |
| johnston-linear-matrix-algebra_FO0360 | 36 | 1.000 | (-A=(-1) A) | | |
| johnston-linear-matrix-algebra_FO0361 | 36 | 0.755 | A=\left[\begin{array}{cc}1 & 3 \\ 2 & -1\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0362 | 36 | 0.755 | C=\left[\begin{array}{ccc}1 & 0 & 1 \\ 0 & -1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0363 | 36 | 1.000 | 2 A-3 B | | |
| johnston-linear-matrix-algebra_FO0364 | 36 | 1.000 | A+2 C | | |
| johnston-linear-matrix-algebra_FO0365 | 37 | 1.000 | A+B=\left[\begin{array}{cc}1 & 3 \\ 2 & -1\end{array}\right]+\left[\begin{array}{ll}2 & 1 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}3 & 4 \\ 2 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0366 | 37 | 0.994 | 2 A-3 B=\left[\begin{array}{cc}2 & 6 \\ 4 & -2\end{array}\right]-\left[\begin{array}{ll}6 & 3 \\ 0 & 3\end{array}\right]=\left[\begin{array}{cc}-4 & 3 \\ 4 & -5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0367 | 37 | 0.999 | 2 \times 2 | | |
| johnston-linear-matrix-algebra_FO0368 | 37 | 0.999 | 2 C | | |
| johnston-linear-matrix-algebra_FO0369 | 37 | 0.991 | 2 \times 3 | | |
| johnston-linear-matrix-algebra_FO0370 | 37 | 1.000 | A, B, C \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO0371 | 37 | 1.000 | A+B=B+A | | |
| johnston-linear-matrix-algebra_FO0372 | 37 | 1.000 | (A+B)+C=A+(B+C) | | |
| johnston-linear-matrix-algebra_FO0373 | 37 | 1.000 | c(A+B)=c A+c B | | |
| johnston-linear-matrix-algebra_FO0374 | 37 | 1.000 | (c+d) A=c A+d A | | |
| johnston-linear-matrix-algebra_FO0375 | 37 | 1.000 | c(d A)=(c d) A | | |
| johnston-linear-matrix-algebra_FO0376 | 37 | 1.000 | a+b=b+a | | |
| johnston-linear-matrix-algebra_FO0377 | 37 | 1.000 | a, b \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0378 | 37 | 1.000 | m n | | |
| johnston-linear-matrix-algebra_FO0379 | 37 | 1.000 | A+O=A | | |
| johnston-linear-matrix-algebra_FO0380 | 37 | 1.000 | A-A=O | | |
| johnston-linear-matrix-algebra_FO0381 | 37 | 1.000 | A \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO0382 | 38 | 1.000 | r \times p | | |
| johnston-linear-matrix-algebra_FO0383 | 38 | 0.910 | n=r | | |
| johnston-linear-matrix-algebra_FO0384 | 38 | 1.000 | B \in \mathcal{M}_{n, p} | | |
| johnston-linear-matrix-algebra_FO0385 | 38 | 1.000 | A B | | |
| johnston-linear-matrix-algebra_FO0386 | 38 | 0.918 | m \times p | | |
| johnston-linear-matrix-algebra_FO0387 | 38 | 0.918 | 1 \leq j \leq p | | |
| johnston-linear-matrix-algebra_FO0389 | 38 | 0.554 | 3 \times 2 | | |
| johnston-linear-matrix-algebra_FO0391 | 38 | 1.000 | A C | | |
| johnston-linear-matrix-algebra_FO0392 | 38 | 1.000 | B A | | |
| johnston-linear-matrix-algebra_FO0393 | 38 | 1.000 | B C | | |
| johnston-linear-matrix-algebra_FO0394 | 38 | 1.000 | B:(1,2) \cdot(5,8)=1 \cdot 5+2 \cdot 8=21 | | |
| johnston-linear-matrix-algebra_FO0395 | 39 | 0.998 | A B^{\prime} | | |
| johnston-linear-matrix-algebra_FO0396 | 39 | 1.000 | n \times p | | |
| johnston-linear-matrix-algebra_FO0397 | 39 | 1.000 | C | | |
| johnston-linear-matrix-algebra_FO0398 | 39 | 1.000 | C:(5,6,7) \cdot(1,0,2)=5 \cdot 1+6 \cdot 0+7 \cdot 2=19 | | |
| johnston-linear-matrix-algebra_FO0399 | 40 | 1.000 | A, B | | |
| johnston-linear-matrix-algebra_FO0400 | 40 | 1.000 | (A B) C=A(B C) | | |
| johnston-linear-matrix-algebra_FO0401 | 40 | 1.000 | A(B+C)=A B+A C | | |
| johnston-linear-matrix-algebra_FO0402 | 40 | 1.000 | (A+B) C=A C+B C | | |
| johnston-linear-matrix-algebra_FO0403 | 40 | 1.000 | c(A B)=(c A) B | | |
| johnston-linear-matrix-algebra_FO0404 | 40 | 0.619 | A(B+C) | | |
| johnston-linear-matrix-algebra_FO0405 | 40 | 0.998 | A B+A C | | |
| johnston-linear-matrix-algebra_FO0406 | 40 | 1.000 | A B= | | |
| johnston-linear-matrix-algebra_FO0407 | 40 | 1.000 | I_{n} | | |
| johnston-linear-matrix-algebra_FO0408 | 40 | 1.000 | I | | |
| johnston-linear-matrix-algebra_FO0409 | 40 | 1.000 | \mathcal{M}_{2} | | |
| johnston-linear-matrix-algebra_FO0410 | 40 | 1.000 | \mathcal{M}_{3} | | |
| johnston-linear-matrix-algebra_FO0411 | 41 | 1.000 | x \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0412 | 41 | 1.000 | x^{0}=1 | | |
| johnston-linear-matrix-algebra_FO0413 | 41 | 1.000 | A^{0}=I | | |
| johnston-linear-matrix-algebra_FO0414 | 41 | 1.000 | A I_{n}=A=I_{m} A | | |
| johnston-linear-matrix-algebra_FO0415 | 41 | 0.996 | A I_{n} | | |
| johnston-linear-matrix-algebra_FO0416 | 41 | 1.000 | A I_{n}=A | | |
| johnston-linear-matrix-algebra_FO0417 | 41 | 1.000 | A^{2}=A A, A^{3}=A A A | | |
| johnston-linear-matrix-algebra_FO0418 | 41 | 1.000 | A^{k+\ell}=A^{k} A^{\ell} | | |
| johnston-linear-matrix-algebra_FO0419 | 41 | 1.000 | \left(A^{k}\right)^{\ell}=A^{k \ell} | | |
| johnston-linear-matrix-algebra_FO0420 | 41 | 1.000 | k, \ell \geq 1 | | |
| johnston-linear-matrix-algebra_FO0421 | 41 | 1.000 | k=0 | | |
| johnston-linear-matrix-algebra_FO0422 | 41 | 1.000 | \ell=0 | | |
| johnston-linear-matrix-algebra_FO0423 | 41 | 1.000 | A=\left[\begin{array}{cc}2 & 1 \\ -1 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0424 | 41 | 1.000 | A^{2} | | |
| johnston-linear-matrix-algebra_FO0425 | 41 | 1.000 | A^{4} | | |
| johnston-linear-matrix-algebra_FO0426 | 41 | 1.000 | I^{7} | | |
| johnston-linear-matrix-algebra_FO0427 | 41 | 1.000 | A^{4}=\left(A^{2}\right)^{2} | | |
| johnston-linear-matrix-algebra_FO0428 | 41 | 1.000 | I^{k}=I | | |
| johnston-linear-matrix-algebra_FO0429 | 41 | 1.000 | k | | |
| johnston-linear-matrix-algebra_FO0430 | 41 | 1.000 | I^{7}=I | | |
| johnston-linear-matrix-algebra_FO0431 | 41 | 1.000 | (A+B)^{2} | | |
| johnston-linear-matrix-algebra_FO0432 | 41 | 1.000 | (A+B)^{2}=A^{2}+2 A B+B^{2} | | |
| johnston-linear-matrix-algebra_FO0433 | 42 | 1.000 | (A+B)^{2}= | | |
| johnston-linear-matrix-algebra_FO0434 | 42 | 1.000 | A^{2}+2 A B+B^{2} | | |
| johnston-linear-matrix-algebra_FO0435 | 42 | 1.000 | (A+I)^{2}=A^{2}+2 A+I | | |
| johnston-linear-matrix-algebra_FO0436 | 42 | 0.994 | 1 \times n | | |
| johnston-linear-matrix-algebra_FO0437 | 42 | 1.000 | m \times 1 | | |
| johnston-linear-matrix-algebra_FO0438 | 42 | 1.000 | A, B \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO0439 | 42 | 1.000 | (A+I)^{2}(A-I) | | |
| johnston-linear-matrix-algebra_FO0440 | 42 | 1.000 | (A-A)(A+B-I)^{7} | | |
| johnston-linear-matrix-algebra_FO0441 | 42 | 1.000 | A+I | | |
| johnston-linear-matrix-algebra_FO0442 | 42 | 1.000 | A-I | | |
| johnston-linear-matrix-algebra_FO0443 | 42 | 1.000 | (A+B-I)^{7} | | |
| johnston-linear-matrix-algebra_FO0444 | 42 | 1.000 | (A-A)(A+ | | |
| johnston-linear-matrix-algebra_FO0445 | 42 | 0.950 | B-I)^{7}=O(A+B-I)^{7}=O | | |
| johnston-linear-matrix-algebra_FO0446 | 42 | 1.000 | \mathbf{v} \in \mathcal{M}_{n, 1} | | |
| johnston-linear-matrix-algebra_FO0447 | 42 | 1.000 | A \mathbf{v} \in \mathcal{M}_{m, 1} | | |
| johnston-linear-matrix-algebra_FO0448 | 42 | 0.942 | A \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0449 | 42 | 1.000 | \mathbf{v}=(2,-1) | | |
| johnston-linear-matrix-algebra_FO0450 | 42 | 1.000 | A \mathbf{v}=(0,2) | | |
| johnston-linear-matrix-algebra_FO0451 | 43 | 1.000 | [1,2,3] \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0452 | 43 | 1.000 | \mathcal{M}_{1,3} | | |
| johnston-linear-matrix-algebra_FO0453 | 43 | 1.000 | \mathbf{v}=[1,2,3] \in \mathcal{M}_{1,3} | | |
| johnston-linear-matrix-algebra_FO0454 | 43 | 1.000 | \mathbf{x}=(1,2,3) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0455 | 43 | 1.000 | \mathbf{v} \mapsto A \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0456 | 43 | 0.774 | A^{T} | | |
| johnston-linear-matrix-algebra_FO0457 | 43 | 0.774 | n \times m | | |
| johnston-linear-matrix-algebra_FO0458 | 43 | 0.774 | a_{j, i} | | |
| johnston-linear-matrix-algebra_FO0459 | 43 | 1.000 | B^{T} | | |
| johnston-linear-matrix-algebra_FO0460 | 43 | 1.000 | (A B)^{T} | | |
| johnston-linear-matrix-algebra_FO0461 | 43 | 1.000 | B^{T} A^{T} | | |
| johnston-linear-matrix-algebra_FO0462 | 43 | 1.000 | A^{T}=\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0463 | 44 | 1.000 | \left[(A B)^{T}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO0464 | 44 | 0.994 | \mathbf{v}^{T} \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0465 | 44 | 0.994 | 1 \times 1 | | |
| johnston-linear-matrix-algebra_FO0466 | 44 | 1.000 | B^{T}=\left[\begin{array}{cc}-1 & 0 \\ 1 & 1 \\ 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0467 | 44 | 0.971 | (A B)^{T}=\left(\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\left[\begin{array}{ccc}-1 & 1 & 1 \\ 0 & 1 & 0\end{array}\right]\right)^{T}=\left[\begin{array}{lll}-1 & 3 & 1 \\ -3 & 7 & 3\end{array}\right]^{T}=\left[\begin{array}{cc}-1 & -3 \\ 3 & 7 \\ 1 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0468 | 44 | 1.000 | B^{T} A^{T}=\left[\begin{array}{cc}-1 & 0 \\ 1 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{cc}-1 & -3 \\ 3 & 7 \\ 1 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0469 | 44 | 1.000 | (A B)^{T}=B^{T} A^{T} | | |
| johnston-linear-matrix-algebra_FO0470 | 44 | 1.000 | \left(A^{T}\right)^{T}=A | | |
| johnston-linear-matrix-algebra_FO0471 | 44 | 1.000 | (A+B)^{T}=A^{T}+B^{T} | | |
| johnston-linear-matrix-algebra_FO0472 | 44 | 1.000 | (c A)^{T}=c A^{T} | | |
| johnston-linear-matrix-algebra_FO0473 | 44 | 1.000 | \left[(A B)^{T}\right]_{i, j}=\left[B^{T} A^{T}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO0474 | 45 | 1.000 | C D | | |
| johnston-linear-matrix-algebra_FO0475 | 45 | 1.000 | D C | | |
| johnston-linear-matrix-algebra_FO0476 | 45 | 0.722 | I_{3} | | |
| johnston-linear-matrix-algebra_FO0477 | 45 | 0.722 | 3 \times 3 | | |
| johnston-linear-matrix-algebra_FO0478 | 45 | 1.000 | D | | |
| johnston-linear-matrix-algebra_FO0479 | 45 | 1.000 | 4 \cdot 5=20 | | |
| johnston-linear-matrix-algebra_FO0480 | 46 | 1.000 | I_{2} | | |
| johnston-linear-matrix-algebra_FO0481 | 46 | 0.926 | I_{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0482 | 46 | 1.000 | C D=\left[\begin{array}{cc}5 & 4 \\ -7 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0483 | 46 | 0.996 | \left[\begin{array}{ll}A & B \\ B & A\end{array}\right]^{2} | | |
| johnston-linear-matrix-algebra_FO0484 | 46 | 0.863 | \left[\begin{array}{ll}A & B \\ O & I_{3}\end{array}\right]\left[\begin{array}{ll}A & A \\ O & A \\ I_{2} & O\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0485 | 46 | 0.434 | \left[\begin{array}{ll}A & B \\ O & I_{3}\end{array}\right]\left[\begin{array}{ll}A & A \\ O & A\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0486 | 46 | 0.995 | \left[\begin{array}{ll}A & B \\ O & I_{3}\end{array}\right]\left[\begin{array}{ll}B & O \\ I_{3} & I_{3}\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0487 | 46 | 0.969 | \left[\begin{array}{ll}A & B \\ B & A\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0488 | 47 | 1.000 | A B+B | | |
| johnston-linear-matrix-algebra_FO0489 | 47 | 1.000 | \mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n} | | |
| johnston-linear-matrix-algebra_FO0490 | 47 | 0.813 | n \times 1 | | |
| johnston-linear-matrix-algebra_FO0491 | 47 | 1.000 | \mathbf{b}_{1}, \mathbf{b}_{2}, \ldots, \mathbf{b}_{p} | | |
| johnston-linear-matrix-algebra_FO0492 | 47 | 1.000 | 1 \times p | | |
| johnston-linear-matrix-algebra_FO0493 | 48 | 1.000 | A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0494 | 48 | 1.000 | B=\left[\begin{array}{ccc}-1 & 1 & 1 \\ 0 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0495 | 48 | 1.000 | 2 B-3 A | | |
| johnston-linear-matrix-algebra_FO0496 | 48 | 0.881 | A-B | | |
| johnston-linear-matrix-algebra_FO0497 | 48 | 1.000 | 2(A+B)-A | | |
| johnston-linear-matrix-algebra_FO0498 | 48 | 1.000 | C D^{T} | | |
| johnston-linear-matrix-algebra_FO0499 | 48 | 1.000 | C^{T} D | | |
| johnston-linear-matrix-algebra_FO0500 | 48 | 1.000 | \left(C^{T} D\right)^{2} | | |
| johnston-linear-matrix-algebra_FO0501 | 48 | 1.000 | (C-D)^{T}\left(A-B^{T}\right) | | |
| johnston-linear-matrix-algebra_FO0502 | 48 | 0.997 | (A+B)(C+D) | | |
| johnston-linear-matrix-algebra_FO0503 | 48 | 1.000 | D^{T}\left(A^{T}+B\right)^{T} C | | |
| johnston-linear-matrix-algebra_FO0504 | 49 | 1.000 | A^{T} A | | |
| johnston-linear-matrix-algebra_FO0505 | 49 | 1.000 | A A^{T} | | |
| johnston-linear-matrix-algebra_FO0506 | 49 | 0.749 | (A B)^{2}=A^{2} B^{2} | | |
| johnston-linear-matrix-algebra_FO0507 | 49 | 1.000 | A B=O | | |
| johnston-linear-matrix-algebra_FO0508 | 49 | 1.000 | A \neq O | | |
| johnston-linear-matrix-algebra_FO0509 | 49 | 1.000 | B=O | | |
| johnston-linear-matrix-algebra_FO0510 | 49 | 0.998 | A^{2}=I | | |
| johnston-linear-matrix-algebra_FO0511 | 49 | 0.998 | A=I | | |
| johnston-linear-matrix-algebra_FO0512 | 49 | 1.000 | A=-I | | |
| johnston-linear-matrix-algebra_FO0513 | 49 | 1.000 | A^{2}=A | | |
| johnston-linear-matrix-algebra_FO0514 | 49 | 1.000 | A=O | | |
| johnston-linear-matrix-algebra_FO0515 | 49 | 1.000 | A \in \mathcal{M}_{n, m} | | |
| johnston-linear-matrix-algebra_FO0516 | 49 | 1.000 | \mathbf{b} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0517 | 49 | 1.000 | A \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO0518 | 49 | 1.000 | A \in \mathcal{M}_{2,2}, B \in \mathcal{M}_{2,5} | | |
| johnston-linear-matrix-algebra_FO0519 | 49 | 1.000 | C \in \mathcal{M}_{5,2} | | |
| johnston-linear-matrix-algebra_FO0520 | 49 | 1.000 | A^{7} | | |
| johnston-linear-matrix-algebra_FO0521 | 49 | 1.000 | B^{2} | | |
| johnston-linear-matrix-algebra_FO0522 | 49 | 1.000 | B B^{T} | | |
| johnston-linear-matrix-algebra_FO0523 | 49 | 0.999 | A B C | | |
| johnston-linear-matrix-algebra_FO0524 | 49 | 1.000 | B^{T} B+C^{T} C | | |
| johnston-linear-matrix-algebra_FO0525 | 49 | 0.963 | A+B C | | |
| johnston-linear-matrix-algebra_FO0526 | 49 | 1.000 | \left(A+C^{T} C\right)\left(B+C^{T}\right) | | |
| johnston-linear-matrix-algebra_FO0527 | 49 | 0.830 | A=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0528 | 49 | 0.830 | A^{1000} | | |
| johnston-linear-matrix-algebra_FO0529 | 49 | 1.000 | A^{2}, A^{3}, \ldots | | |
| johnston-linear-matrix-algebra_FO0530 | 49 | 1.000 | B=\left[\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0531 | 49 | 1.000 | B^{1000} | | |
| johnston-linear-matrix-algebra_FO0532 | 49 | 1.000 | B^{2}, B^{3}, \ldots | | |
| johnston-linear-matrix-algebra_FO0533 | 49 | 0.972 | A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0534 | 49 | 1.000 | A^{3} | | |
| johnston-linear-matrix-algebra_FO0535 | 49 | 1.000 | A^{k} | | |
| johnston-linear-matrix-algebra_FO0536 | 49 | 1.000 | k \geq 0 | | |
| johnston-linear-matrix-algebra_FO0537 | 49 | 1.000 | B=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0538 | 49 | 1.000 | B^{3} | | |
| johnston-linear-matrix-algebra_FO0539 | 49 | 1.000 | B^{k} | | |
| johnston-linear-matrix-algebra_FO0540 | 49 | 1.000 | C=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0541 | 49 | 1.000 | C^{2} | | |
| johnston-linear-matrix-algebra_FO0542 | 49 | 1.000 | C^{3} | | |
| johnston-linear-matrix-algebra_FO0543 | 49 | 1.000 | C^{k} | | |
| johnston-linear-matrix-algebra_FO0544 | 49 | 0.342 | h \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0545 | 49 | 1.000 | A^{2500} | | |
| johnston-linear-matrix-algebra_FO0546 | 49 | 1.000 | h=0.39 | | |
| johnston-linear-matrix-algebra_FO0547 | 49 | 1.000 | h=0.40 | | |
| johnston-linear-matrix-algebra_FO0548 | 49 | 1.000 | h=0.41 | | |
| johnston-linear-matrix-algebra_FO0549 | 49 | 0.948 | J_{n} | | |
| johnston-linear-matrix-algebra_FO0550 | 49 | 1.000 | J_{n}^{2} | | |
| johnston-linear-matrix-algebra_FO0551 | 49 | 1.000 | A \in \mathcal{M}_{2,2}, B \in \mathcal{M}_{2,3} | | |
| johnston-linear-matrix-algebra_FO0552 | 49 | 1.000 | C \in \mathcal{M}_{3,4} | | |
| johnston-linear-matrix-algebra_FO0553 | 49 | 0.885 | \left[\begin{array}{cc}A & A \\ A & A\end{array}\right]^{2} | | |
| johnston-linear-matrix-algebra_FO0554 | 49 | 0.999 | \left[\begin{array}{cc}C & I_{3} \\ I_{4} & O\end{array}\right]\left[\begin{array}{cc}O & I_{3} \\ I_{4} & C^{T}\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0555 | 49 | 1.000 | \left[\begin{array}{cc}A & B \\ B^{T} & I_{3}\end{array}\right]\left[\begin{array}{cc}O & C^{T} \\ B^{T} & I_{3}\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0556 | 49 | 1.000 | \left[\begin{array}{cc}A & B \\ O & C^{T}\end{array}\right]\left[\begin{array}{cc}I_{2} & B \\ B^{T} & O\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0557 | 49 | 1.000 | A=\left[\begin{array}{ll|ll}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\end{array}\right], B=\left[\begin{array}{ll|ll}1 & 2 & 2 & 1 \\ 2 & 1 & 1 & 2 \\ \hline 2 & 1 & 1 & 2 \\ 1 & 2 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0558 | 49 | 0.999 | A=\left[\begin{array}{ll|l}1 & 2 & 0 \\ 3 & 1 & 0 \\ \hline 0 & 0 & 2\end{array}\right], B=\left[\begin{array}{cc|cc}-1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ \hline 0 & 0 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0559 | 49 | 1.000 | A=\left[\begin{array}{lll|l}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ \hline 0 & 0 & 0 & 2\end{array}\right], B=\left[\begin{array}{ll|l}1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ \hline 1 & 2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0560 | 50 | 0.991 | 7 \times 7 | | |
| johnston-linear-matrix-algebra_FO0561 | 50 | 0.999 | B \in \mathcal{M}_{r, p} | | |
| johnston-linear-matrix-algebra_FO0562 | 50 | 1.000 | m, n, r | | |
| johnston-linear-matrix-algebra_FO0563 | 50 | 1.000 | p | | |
| johnston-linear-matrix-algebra_FO0564 | 50 | 1.000 | \mathbf{v}^{T} A \mathbf{w}=\mathbf{v}^{T} B \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0565 | 50 | 1.000 | \mathbf{v} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0566 | 50 | 1.000 | \mathbf{w} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0567 | 50 | 1.000 | A=B | | |
| johnston-linear-matrix-algebra_FO0568 | 50 | 1.000 | \mathbf{e}_{i}^{T} A \mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO0569 | 50 | 1.000 | \mathbf{x} \cdot(A \mathbf{y})=\left(A^{T} \mathbf{x}\right) \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO0570 | 50 | 1.000 | \mathbf{x} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0571 | 50 | 1.000 | \mathbf{x} \cdot(A \mathbf{y})=(B \mathbf{x}) \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO0572 | 50 | 1.000 | B=A^{T} | | |
| johnston-linear-matrix-algebra_FO0573 | 50 | 0.919 | \mathbf{e}_{i} | | |
| johnston-linear-matrix-algebra_FO0574 | 50 | 1.000 | A \mathbf{e}_{i} | | |
| johnston-linear-matrix-algebra_FO0575 | 50 | 1.000 | \mathbf{e}_{i}^{T} A | | |
| johnston-linear-matrix-algebra_FO0576 | 50 | 1.000 | A \mathbf{v}=B \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0577 | 50 | 1.000 | I_{m} A=A | | |
| johnston-linear-matrix-algebra_FO0578 | 50 | 1.000 | A \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO0579 | 50 | 1.000 | \ell | | |
| johnston-linear-matrix-algebra_FO0580 | 50 | 1.000 | A_{1}, A_{2}, \ldots, A_{k} | | |
| johnston-linear-matrix-algebra_FO0581 | 50 | 1.000 | A_{1} A_{2} \ldots A_{k} | | |
| johnston-linear-matrix-algebra_FO0582 | 50 | 1.000 | \left(A_{1} A_{2} \cdots A_{k}\right)^{T}=A_{k}^{T} \cdots A_{2}^{T} A_{1}^{T} | | |
| johnston-linear-matrix-algebra_FO0583 | 50 | 1.000 | \left(A^{n}\right)^{T}=\left(A^{T}\right)^{n} | | |
| johnston-linear-matrix-algebra_FO0584 | 50 | 1.000 | n \geq 0 | | |
| johnston-linear-matrix-algebra_FO0585 | 50 | 1.000 | \mathbf{a}_{j}^{T} | | |
| johnston-linear-matrix-algebra_FO0586 | 51 | 1.000 | T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0587 | 51 | 1.000 | T | | |
| johnston-linear-matrix-algebra_FO0588 | 51 | 1.000 | \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0589 | 51 | 1.000 | f: X \rightarrow Y | | |
| johnston-linear-matrix-algebra_FO0590 | 51 | 1.000 | X | | |
| johnston-linear-matrix-algebra_FO0591 | 51 | 1.000 | Y | | |
| johnston-linear-matrix-algebra_FO0592 | 51 | 1.000 | T(\mathbf{0})=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0593 | 51 | 1.000 | T(\mathbf{v}+\mathbf{w})=T(\mathbf{v})+T(\mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0594 | 51 | 1.000 | T(c \mathbf{v})=c T(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0595 | 51 | 1.000 | A \mathbf{v} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0596 | 51 | 1.000 | A(\mathbf{v}+\mathbf{w})=A \mathbf{v}+A \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0597 | 51 | 0.999 | A(c \mathbf{v})=c(A \mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0598 | 51 | 1.000 | T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0599 | 51 | 1.000 | T(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0600 | 51 | 1.000 | T\left(\mathbf{e}_{1}\right) | | |
| johnston-linear-matrix-algebra_FO0601 | 51 | 1.000 | T\left(\mathbf{e}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0602 | 51 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(1+v_{1}, 2+v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0603 | 51 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(v_{1}-v_{2}, v_{1} v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0604 | 51 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(v_{1}-v_{2}, v_{1}+v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0605 | 51 | 1.000 | 2 T(0,0)=2(1,2)=(2,4) | | |
| johnston-linear-matrix-algebra_FO0606 | 51 | 1.000 | T(2(0,0))=T(0,0)=(1,2) | | |
| johnston-linear-matrix-algebra_FO0607 | 52 | 1.000 | 2 T(1,1)=2(0,1)=(0,2) | | |
| johnston-linear-matrix-algebra_FO0608 | 52 | 1.000 | T(2(1,1))=T(2,2)=(0,4) | | |
| johnston-linear-matrix-algebra_FO0609 | 52 | 1.000 | \sqrt{2} | | |
| johnston-linear-matrix-algebra_FO0610 | 53 | 1.000 | \mathbf{v}=v_{1} \mathbf{e}_{1}+v_{2} \mathbf{e}_{2}+\cdots+ | | |
| johnston-linear-matrix-algebra_FO0611 | 53 | 1.000 | v_{n} \mathbf{e}_{n} | | |
| johnston-linear-matrix-algebra_FO0612 | 53 | 1.000 | \mathbf{v} \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0613 | 53 | 1.000 | v_{1} | | |
| johnston-linear-matrix-algebra_FO0614 | 53 | 1.000 | v_{2} | | |
| johnston-linear-matrix-algebra_FO0615 | 53 | 1.000 | \mathbf{e}_{3}, \ldots, \mathbf{e}_{n} | | |
| johnston-linear-matrix-algebra_FO0616 | 53 | 1.000 | T\left(\mathbf{e}_{1}\right), T\left(\mathbf{e}_{2}\right), \ldots, T\left(\mathbf{e}_{n}\right) | | |
| johnston-linear-matrix-algebra_FO0617 | 53 | 1.000 | T\left(\mathbf{e}_{1}\right)=(1,1) | | |
| johnston-linear-matrix-algebra_FO0618 | 53 | 1.000 | T\left(\mathbf{e}_{2}\right)=(-1,1) | | |
| johnston-linear-matrix-algebra_FO0619 | 53 | 1.000 | T(2,3) | | |
| johnston-linear-matrix-algebra_FO0620 | 53 | 1.000 | T\left(v_{1}, v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0621 | 53 | 1.000 | (2,3)=2 \mathbf{e}_{1}+3 \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO0622 | 53 | 1.000 | T(2,3)=T\left(2 \mathbf{e}_{1}+3 \mathbf{e}_{2}\right)= | | |
| johnston-linear-matrix-algebra_FO0623 | 53 | 1.000 | 2 T\left(\mathbf{e}_{1}\right)+3 T\left(\mathbf{e}_{2}\right)=2(1,1)+3(-1,1)=(-1,5) | | |
| johnston-linear-matrix-algebra_FO0624 | 53 | 1.000 | \left(v_{1}, v_{2}\right)=v_{1} \mathbf{e}_{1}+v_{2} \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO0625 | 54 | 1.000 | [T] | | |
| johnston-linear-matrix-algebra_FO0626 | 54 | 1.000 | T\left(\mathbf{e}_{2}\right), \ldots, T\left(\mathbf{e}_{n}\right) | | |
| johnston-linear-matrix-algebra_FO0627 | 54 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(v_{1}-\right. | | |
| johnston-linear-matrix-algebra_FO0628 | 54 | 0.951 | v_{2}, v_{1}+v_{2} | | |
| johnston-linear-matrix-algebra_FO0629 | 54 | 1.000 | [T] \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO0630 | 54 | 0.623 | [T] \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0631 | 54 | 1.000 | [T] \mathbf{v}=T(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0632 | 54 | 1.000 | T(\mathbf{v})=A \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0633 | 54 | 1.000 | T(\mathbf{v})=[T] \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0634 | 54 | 1.000 | [T] \mathbf{v}=A \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0635 | 54 | 1.000 | A=[T] | | |
| johnston-linear-matrix-algebra_FO0636 | 55 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(v_{1}+2 v_{2}, 3 v_{1}+4 v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0637 | 55 | 1.000 | T\left(v_{1}, v_{2}, v_{3}\right)=\left(3 v_{1}-v_{2}+v_{3}, 2 v_{1}+4 v_{2}-2 v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO0638 | 55 | 1.000 | T\left(\mathbf{e}_{1}\right)=(1,3) | | |
| johnston-linear-matrix-algebra_FO0639 | 55 | 1.000 | T\left(\mathbf{e}_{2}\right)=(2,4) | | |
| johnston-linear-matrix-algebra_FO0640 | 55 | 1.000 | T\left(\mathbf{e}_{1}\right), T\left(\mathbf{e}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0641 | 55 | 1.000 | T\left(\mathbf{e}_{3}\right) | | |
| johnston-linear-matrix-algebra_FO0642 | 55 | 1.000 | v_{1}, v_{2} | | |
| johnston-linear-matrix-algebra_FO0643 | 55 | 1.000 | v_{3} | | |
| johnston-linear-matrix-algebra_FO0644 | 55 | 1.000 | A=\left[\begin{array}{ll}2.1 & 0.3 \\ 0.2 & 1.2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0645 | 55 | 1.000 | B=\left[\begin{array}{ll}2 & 1 \\ 1 & 2 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0646 | 55 | 0.508 | A \mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO0647 | 55 | 0.508 | A \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO0648 | 55 | 0.530 | B \mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO0649 | 55 | 0.530 | 2,1,1 | | |
| johnston-linear-matrix-algebra_FO0650 | 55 | 0.530 | B \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO0651 | 55 | 0.637 | (1,2,1) | | |
| johnston-linear-matrix-algebra_FO0652 | 56 | 1.000 | S, T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0653 | 56 | 1.000 | S+T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0654 | 56 | 1.000 | c T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0655 | 56 | 1.000 | S+T | | |
| johnston-linear-matrix-algebra_FO0656 | 56 | 1.000 | [S+T]=[S]+[T] | | |
| johnston-linear-matrix-algebra_FO0657 | 56 | 1.000 | c T | | |
| johnston-linear-matrix-algebra_FO0658 | 56 | 1.000 | [c T]=c[T] | | |
| johnston-linear-matrix-algebra_FO0659 | 56 | 1.000 | O: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0660 | 56 | 1.000 | O(\mathbf{v})=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0661 | 56 | 1.000 | I: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0662 | 56 | 1.000 | I(\mathbf{v})=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0663 | 56 | 1.000 | O \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO0664 | 56 | 1.000 | I \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO0665 | 56 | 1.000 | O \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0666 | 56 | 1.000 | I \mathbf{v}=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0667 | 56 | 1.000 | T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0668 | 56 | 1.000 | c_{1}, c_{2}, \ldots, c_{n} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0669 | 56 | 0.995 | T\left(v_{1}, v_{2}, \ldots, v_{n}\right)=\left(c_{1} v_{1}, c_{2} v_{2}, \ldots, c_{n} v_{n}\right) | | |
| johnston-linear-matrix-algebra_FO0670 | 57 | 1.000 | D A | | |
| johnston-linear-matrix-algebra_FO0671 | 57 | 1.000 | A D | | |
| johnston-linear-matrix-algebra_FO0672 | 57 | 0.999 | \mathbf{v}=\left(v_{1}, v_{2}\right) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0673 | 58 | 1.000 | \mathbf{u}^{T} \mathbf{u} | | |
| johnston-linear-matrix-algebra_FO0674 | 58 | 1.000 | \mathbf{u} \cdot \mathbf{u}=\|\mathbf{u}\|^{2}=1 | | |
| johnston-linear-matrix-algebra_FO0675 | 58 | 1.000 | \mathbf{u u}^{T} | | |
| johnston-linear-matrix-algebra_FO0676 | 58 | 1.000 | \mathbf{u}^{T} | | |
| johnston-linear-matrix-algebra_FO0677 | 58 | 1.000 | 1 \times n, \mathbf{u u}^{T} | | |
| johnston-linear-matrix-algebra_FO0678 | 58 | 1.000 | P_{\mathbf{u}} | | |
| johnston-linear-matrix-algebra_FO0679 | 58 | 0.999 | P_{\mathbf{u}}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0680 | 58 | 0.999 | \|\mathbf{v}\| \cos (\theta) | | |
| johnston-linear-matrix-algebra_FO0681 | 58 | 1.000 | P_{\mathbf{u}}(\mathbf{v})=\mathbf{u}(\|\mathbf{v}\| \cos (\theta)) | | |
| johnston-linear-matrix-algebra_FO0682 | 58 | 1.000 | \cos (\theta)=\mathbf{u} \cdot \mathbf{v} /(\|\mathbf{u}\|\|\mathbf{v}\|) | | |
| johnston-linear-matrix-algebra_FO0683 | 58 | 1.000 | \|\mathbf{u}\|=1 | | |
| johnston-linear-matrix-algebra_FO0684 | 58 | 1.000 | \mathbf{u} \cdot \mathbf{v}=\mathbf{u}^{T} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0685 | 58 | 1.000 | P_{\mathbf{u}}(\mathbf{v})= | | |
| johnston-linear-matrix-algebra_FO0686 | 58 | 1.000 | \mathbf{u}\left(\mathbf{u}^{T} \mathbf{v}\right)=\left(\mathbf{u u}^{T}\right) \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0687 | 58 | 1.000 | \mathbf{u}=(1,0) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0688 | 58 | 1.000 | \mathbf{w}=(1,2,3) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0689 | 59 | 1.000 | \mathbf{u} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0690 | 59 | 1.000 | P_{\mathbf{u}}(\mathbf{u})=\mathbf{u}: P_{\mathbf{u}} | | |
| johnston-linear-matrix-algebra_FO0691 | 59 | 1.000 | F | | |
| johnston-linear-matrix-algebra_FO0692 | 59 | 1.000 | F_{\mathbf{u}} | | |
| johnston-linear-matrix-algebra_FO0693 | 59 | 1.000 | \left[P_{\mathbf{u}}\right] \mathbf{e}_{1}=\mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO0694 | 59 | 1.000 | \left[P_{\mathbf{u}}\right] \mathbf{e}_{2}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0695 | 59 | 1.000 | \|\mathbf{w}\|=\sqrt{1^{2}+2^{2}+3^{2}}=\sqrt{14} | | |
| johnston-linear-matrix-algebra_FO0696 | 59 | 1.000 | \mathbf{u}= | | |
| johnston-linear-matrix-algebra_FO0697 | 59 | 0.987 | \mathbf{w} /\|\mathbf{w}\|=(1,2,3) / \sqrt{14} | | |
| johnston-linear-matrix-algebra_FO0698 | 59 | 1.000 | \mathbf{e}_{1}, \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO0699 | 59 | 1.000 | \mathbf{e}_{3} | | |
| johnston-linear-matrix-algebra_FO0700 | 59 | 0.999 | F_{\mathbf{u}}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0701 | 59 | 1.000 | F_{\mathbf{u}}(\mathbf{v})=\mathbf{v}+2\left(P_{\mathbf{u}}(\mathbf{v})-\mathbf{v}\right) | | |
| johnston-linear-matrix-algebra_FO0702 | 60 | 1.000 | F_{\mathbf{u}}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0704 | 60 | 1.000 | \mathbf{u}=(0,1) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0705 | 60 | 1.000 | \mathbf{w}=(1,1,1) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0706 | 60 | 1.000 | \left[F_{\mathbf{u}}\right] | | |
| johnston-linear-matrix-algebra_FO0707 | 61 | 1.000 | F_{\mathbf{u}}(\mathbf{u})=\mathbf{u}: F_{\mathbf{u}} | | |
| johnston-linear-matrix-algebra_FO0708 | 61 | 1.000 | \|\mathbf{w}\|=\sqrt{1^{2}+1^{2}+1^{2}}=\sqrt{3} | | |
| johnston-linear-matrix-algebra_FO0709 | 61 | 1.000 | \mathbf{u}=\mathbf{w} /\|\mathbf{w}\|= | | |
| johnston-linear-matrix-algebra_FO0710 | 61 | 0.999 | (1,1,1) / \sqrt{3} | | |
| johnston-linear-matrix-algebra_FO0711 | 61 | 1.000 | \mathbf{v}=(-1,3) | | |
| johnston-linear-matrix-algebra_FO0712 | 61 | 1.000 | \pi / 3 | | |
| johnston-linear-matrix-algebra_FO0713 | 62 | 1.000 | R^{\theta} | | |
| johnston-linear-matrix-algebra_FO0714 | 62 | 1.000 | \mathbf{u}=(\cos (\pi / 3), \sin (\pi / 3))=(1, \sqrt{3}) / 2 | | |
| johnston-linear-matrix-algebra_FO0715 | 62 | 1.000 | R^{\theta}: \mathbb{R}^{2} \rightarrow | | |
| johnston-linear-matrix-algebra_FO0716 | 62 | 1.000 | R^{\theta}(\mathbf{0})=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0717 | 62 | 1.000 | R^{\theta}(\mathbf{v}+\mathbf{w})=R^{\theta}(\mathbf{v})+R^{\theta}(\mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0718 | 62 | 1.000 | R^{\theta}(c \mathbf{v})=c R^{\theta}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0719 | 62 | 1.000 | R^{\theta}\left(\mathbf{e}_{1}\right) | | |
| johnston-linear-matrix-algebra_FO0720 | 63 | 1.000 | \sin (-x)=-\sin (x) | | |
| johnston-linear-matrix-algebra_FO0721 | 63 | 1.000 | \cos (-x)=\cos (x) | | |
| johnston-linear-matrix-algebra_FO0722 | 63 | 1.000 | R^{\theta}\left(\mathbf{e}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0723 | 63 | 1.000 | R^{\theta}\left(\mathbf{e}_{1}\right)=(\cos (\theta), \sin (\theta)) | | |
| johnston-linear-matrix-algebra_FO0724 | 63 | 1.000 | R^{\theta}\left(\mathbf{e}_{2}\right)=(-\sin (\theta), \cos (\theta)) | | |
| johnston-linear-matrix-algebra_FO0725 | 63 | 1.000 | \pi / 4 | | |
| johnston-linear-matrix-algebra_FO0726 | 63 | 1.000 | \pi / 6 | | |
| johnston-linear-matrix-algebra_FO0727 | 63 | 1.000 | -\pi / 6 | | |
| johnston-linear-matrix-algebra_FO0728 | 63 | 1.000 | \mathbf{v}=(1,3) | | |
| johnston-linear-matrix-algebra_FO0729 | 63 | 1.000 | \mathbf{w}=(\sqrt{3}, 3) | | |
| johnston-linear-matrix-algebra_FO0730 | 64 | 1.000 | R^{\pi / 6}(\mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0731 | 64 | 1.000 | \mathbf{v}=(1,2,3) | | |
| johnston-linear-matrix-algebra_FO0732 | 64 | 1.000 | R_{y z}^{\theta} | | |
| johnston-linear-matrix-algebra_FO0733 | 65 | 1.000 | \left[R_{z x}^{\theta}\right] | | |
| johnston-linear-matrix-algebra_FO0734 | 65 | 1.000 | \left[R_{x z}^{\theta}\right] | | |
| johnston-linear-matrix-algebra_FO0735 | 65 | 1.000 | -\sin (\theta) | | |
| johnston-linear-matrix-algebra_FO0736 | 65 | 1.000 | R_{y z}^{\theta}\left(\mathbf{e}_{1}\right)=\mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO0737 | 65 | 1.000 | y z | | |
| johnston-linear-matrix-algebra_FO0738 | 65 | 1.000 | R_{y z}^{\theta}\left(\mathbf{e}_{2}\right)=(0, \cos (\theta), \sin (\theta)) | | |
| johnston-linear-matrix-algebra_FO0739 | 65 | 0.551 | R_{y z}^{\theta}\left(\mathbf{e}_{3}\right)=(0,-\sin (\theta), \cos (\theta)) | | |
| johnston-linear-matrix-algebra_FO0740 | 65 | 0.999 | \mathbf{v}=(3,-1,2) | | |
| johnston-linear-matrix-algebra_FO0741 | 65 | 0.999 | \theta=2 \pi / 3 | | |
| johnston-linear-matrix-algebra_FO0742 | 65 | 1.000 | R_{x y}^{2 \pi / 3}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0743 | 65 | 1.000 | R_{x y}^{2 \pi / 3} | | |
| johnston-linear-matrix-algebra_FO0744 | 66 | 1.000 | S \circ T | | |
| johnston-linear-matrix-algebra_FO0745 | 66 | 0.907 | (S \circ T)(\mathbf{v})=S(T(\mathbf{v})) | | |
| johnston-linear-matrix-algebra_FO0746 | 66 | 0.999 | S, T: \mathbb{R}^{n} \rightarrow | | |
| johnston-linear-matrix-algebra_FO0747 | 66 | 1.000 | S+c T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0748 | 66 | 1.000 | S: \mathbb{R}^{m} \rightarrow \mathbb{R}^{p} | | |
| johnston-linear-matrix-algebra_FO0749 | 66 | 1.000 | S \circ T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{p} | | |
| johnston-linear-matrix-algebra_FO0750 | 66 | 1.000 | \mathbb{R}^{p} | | |
| johnston-linear-matrix-algebra_FO0751 | 67 | 1.000 | [S \circ T]=[S][T] | | |
| johnston-linear-matrix-algebra_FO0752 | 67 | 1.000 | p \times n | | |
| johnston-linear-matrix-algebra_FO0753 | 67 | 1.000 | [D]\left[F_{\mathbf{u}}\right] | | |
| johnston-linear-matrix-algebra_FO0754 | 67 | 0.980 | \operatorname{not}\left[F_{\mathbf{u}}\right][D] | | |
| johnston-linear-matrix-algebra_FO0756 | 67 | 1.000 | [S] \in \mathcal{M}_{p, m} | | |
| johnston-linear-matrix-algebra_FO0757 | 67 | 1.000 | \mathbb{R}^{n} \rightarrow \mathbb{R}^{p} | | |
| johnston-linear-matrix-algebra_FO0758 | 67 | 1.000 | (S \circ T)(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0759 | 67 | 1.000 | [S][T] | | |
| johnston-linear-matrix-algebra_FO0760 | 67 | 1.000 | y=\frac{4}{3} x | | |
| johnston-linear-matrix-algebra_FO0761 | 67 | 0.960 | \mathbf{u}=(3 / 5,4 / 5) | | |
| johnston-linear-matrix-algebra_FO0762 | 67 | 1.000 | T=D \circ F_{\mathbf{u}} | | |
| johnston-linear-matrix-algebra_FO0763 | 68 | 0.999 | \phi | | |
| johnston-linear-matrix-algebra_FO0764 | 68 | 0.999 | R^{\theta} \circ R^{\phi}=R^{\theta+\phi} | | |
| johnston-linear-matrix-algebra_FO0765 | 69 | 1.000 | \phi=\theta | | |
| johnston-linear-matrix-algebra_FO0766 | 69 | 1.000 | \sin (2 \theta)= | | |
| johnston-linear-matrix-algebra_FO0767 | 69 | 1.000 | 2 \sin (\theta) \cos (\theta) | | |
| johnston-linear-matrix-algebra_FO0768 | 69 | 1.000 | \cos (2 \theta)= | | |
| johnston-linear-matrix-algebra_FO0769 | 69 | 0.980 | \cos ^{2}(\theta)-\sin ^{2}(\theta) | | |
| johnston-linear-matrix-algebra_FO0770 | 69 | 0.995 | \theta+\phi | | |
| johnston-linear-matrix-algebra_FO0771 | 69 | 1.000 | R^{\theta} \circ R^{\phi} | | |
| johnston-linear-matrix-algebra_FO0772 | 69 | 1.000 | R^{\theta+\phi} | | |
| johnston-linear-matrix-algebra_FO0773 | 69 | 1.000 | \left[R^{\theta} \circ R^{\phi}\right] | | |
| johnston-linear-matrix-algebra_FO0774 | 69 | 1.000 | \left[R^{\theta+\phi}\right] | | |
| johnston-linear-matrix-algebra_FO0775 | 69 | 0.983 | (2,1) | | |
| johnston-linear-matrix-algebra_FO0776 | 69 | 0.831 | \left[R^{\theta}\right] | | |
| johnston-linear-matrix-algebra_FO0777 | 69 | 1.000 | R | | |
| johnston-linear-matrix-algebra_FO0778 | 70 | 1.000 | R, S | | |
| johnston-linear-matrix-algebra_FO0779 | 70 | 1.000 | R \circ S \circ T | | |
| johnston-linear-matrix-algebra_FO0780 | 70 | 1.000 | [R \circ S \circ T]=[R][S][T] | | |
| johnston-linear-matrix-algebra_FO0781 | 70 | 0.999 | T\left(v_{1}, v_{2}\right)=\left(v_{1}+2 v_{2}, 3 v_{1}-v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0782 | 70 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(v_{1}+v_{2}, 2 v_{1}-v_{2},-v_{1}+3 v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0783 | 70 | 0.924 | T\left(v_{1}, v_{2}, v_{3}\right)=\left(v_{1}+v_{2}, v_{1}+v_{2}-v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO0784 | 70 | 1.000 | T\left(v_{1}, v_{2}, v_{3}\right)=\left(v_{2}, 2 v_{1}+v_{3}, v_{2}-v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO0785 | 70 | 0.669 | T(1,0)=(3,-1) | | |
| johnston-linear-matrix-algebra_FO0786 | 70 | 0.669 | T(0,1)=(1,2) | | |
| johnston-linear-matrix-algebra_FO0787 | 70 | 1.000 | T(1,0)=(1,3) | | |
| johnston-linear-matrix-algebra_FO0788 | 70 | 1.000 | T(1,1)=(3,7) | | |
| johnston-linear-matrix-algebra_FO0789 | 70 | 0.971 | T(1,0)=(-1,0,1) | | |
| johnston-linear-matrix-algebra_FO0790 | 70 | 0.971 | T(0,1)=(2,3,0) | | |
| johnston-linear-matrix-algebra_FO0791 | 70 | 0.975 | T(1,0,0)=(2,1), \quad T(0,1,0)=(-1,1) | | |
| johnston-linear-matrix-algebra_FO0792 | 70 | 1.000 | T(0,0,1)=(0,3) | | |
| johnston-linear-matrix-algebra_FO0793 | 70 | 0.539 | T(1,0,0)=(1,2,3), T(1,1,0)=(0,1,2) | | |
| johnston-linear-matrix-algebra_FO0794 | 70 | 0.976 | T(1,1,1)=(0,0,1) | | |
| johnston-linear-matrix-algebra_FO0795 | 70 | 1.000 | T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{4} | | |
| johnston-linear-matrix-algebra_FO0796 | 70 | 1.000 | \mathbb{R}^{4} \rightarrow \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0797 | 70 | 1.000 | 4 \times 3 | | |
| johnston-linear-matrix-algebra_FO0798 | 70 | 0.999 | T: \mathbb{R}^{2} \rightarrow | | |
| johnston-linear-matrix-algebra_FO0799 | 70 | 1.000 | T\left(\mathbf{e}_{1}\right)=(2,1), T\left(\mathbf{e}_{2}\right)=(1,3) | | |
| johnston-linear-matrix-algebra_FO0800 | 70 | 1.000 | T(1,1)=(3,3) | | |
| johnston-linear-matrix-algebra_FO0801 | 70 | 0.966 | *(\mathbf{g}) | | |
| johnston-linear-matrix-algebra_FO0802 | 70 | 0.966 | R_{x y}^{\theta}, R_{y z}^{\theta} | | |
| johnston-linear-matrix-algebra_FO0803 | 70 | 0.966 | R_{x z}^{\theta} | | |
| johnston-linear-matrix-algebra_FO0804 | 70 | 0.860 | R_{x y}^{\theta} \circ R_{y z}^{\theta}=R_{x z}^{\theta} | | |
| johnston-linear-matrix-algebra_FO0805 | 70 | 1.000 | \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0806 | 70 | 0.972 | T\left(v_{1}, v_{2}\right)=\left(v_{1}^{2}, v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0807 | 70 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(v_{1}+2 v_{2}, v_{2}-v_{1}\right) | | |
| johnston-linear-matrix-algebra_FO0808 | 70 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(\sin \left(v_{1}\right)+v_{2}, v_{1}-\cos \left(v_{2}\right)\right) | | |
| johnston-linear-matrix-algebra_FO0809 | 70 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(\sin (3) v_{1}+v_{2}, v_{1}-\cos (2) v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0810 | 70 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(\sqrt{v_{1}}+\sqrt{v_{2}}, \sqrt{v_{1}+v_{2}}\right) | | |
| johnston-linear-matrix-algebra_FO0811 | 70 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(\sqrt{3} v_{1}, \sqrt{2} v_{2}-\sqrt{5} v_{1}\right) | | |
| johnston-linear-matrix-algebra_FO0812 | 70 | 0.993 | T\left(v_{1}, v_{2}\right)=\left(\left|v_{1}\right|,\left|v_{2}\right|\right) | | |
| johnston-linear-matrix-algebra_FO0813 | 70 | 1.000 | y=3 x | | |
| johnston-linear-matrix-algebra_FO0814 | 70 | 1.000 | y=3 x-2 | | |
| johnston-linear-matrix-algebra_FO0815 | 70 | 1.000 | y=2 x | | |
| johnston-linear-matrix-algebra_FO0816 | 70 | 1.000 | y=2 x+1 | | |
| johnston-linear-matrix-algebra_FO0817 | 70 | 0.999 | \pi / 5 | | |
| johnston-linear-matrix-algebra_FO0818 | 70 | 1.000 | (0,1) | | |
| johnston-linear-matrix-algebra_FO0819 | 70 | 0.981 | S\left(v_{1}, v_{2}\right)=\left(2 v_{2}, v_{1}+v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0820 | 70 | 1.000 | S\left(v_{1}, v_{2}\right)=\left(v_{1}-2 v_{2}, 3 v_{1}+v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO0821 | 70 | 0.999 | S\left(v_{1}, v_{2}, v_{3}\right)=\left(v_{1}, v_{1}+v_{2}, v_{1}+v_{2}+v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO0822 | 70 | 0.999 | y=x | | |
| johnston-linear-matrix-algebra_FO0823 | 70 | 1.000 | y=x / 2 | | |
| johnston-linear-matrix-algebra_FO0824 | 70 | 1.000 | \mathbf{v}_{1}, \ldots, \mathbf{v}_{k} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0825 | 70 | 1.000 | c_{1}, \ldots, c_{k} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0826 | 70 | 1.000 | D \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO0827 | 70 | 1.000 | 1 \leq i \leq n, \ldots | | |
| johnston-linear-matrix-algebra_FO0828 | 70 | 1.000 | d_{i, i} | | |
| johnston-linear-matrix-algebra_FO0829 | 71 | 1.000 | Q, S, T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO0830 | 71 | 1.000 | S+T=T+S | | |
| johnston-linear-matrix-algebra_FO0831 | 71 | 1.000 | Q+(S+T)=(Q+S)+T | | |
| johnston-linear-matrix-algebra_FO0832 | 71 | 1.000 | c(S+T)=c S+c T | | |
| johnston-linear-matrix-algebra_FO0833 | 71 | 1.000 | (c+d) T=c T+d T | | |
| johnston-linear-matrix-algebra_FO0834 | 71 | 1.000 | c(d T)=(c d) T | | |
| johnston-linear-matrix-algebra_FO0835 | 71 | 1.000 | A^{160} | | |
| johnston-linear-matrix-algebra_FO0836 | 71 | 1.000 | \theta \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO0837 | 71 | 1.000 | A^{T} A=I | | |
| johnston-linear-matrix-algebra_FO0838 | 71 | 0.999 | A=\mathbf{u u}^{T} \in | | |
| johnston-linear-matrix-algebra_FO0839 | 71 | 1.000 | A= | | |
| johnston-linear-matrix-algebra_FO0840 | 71 | 0.481 | 2 \mathbf{u u}^{T}-I \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO0841 | 71 | 1.000 | P_{\mathbf{u}}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0842 | 71 | 1.000 | \left\|P_{\mathbf{u}}(\mathbf{v})\right\| \leq\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0843 | 71 | 1.000 | y=m x | | |
| johnston-linear-matrix-algebra_FO0844 | 71 | 1.000 | \mathbf{u}, \mathbf{v} \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO0845 | 71 | 1.000 | F_{\mathbf{u}} \circ F_{\mathbf{v}}=R^{2 \theta} | | |
| johnston-linear-matrix-algebra_FO0846 | 71 | 1.000 | S_{i, j}^{c} | | |
| johnston-linear-matrix-algebra_FO0847 | 71 | 1.000 | S_{1,2}^{c} | | |
| johnston-linear-matrix-algebra_FO0848 | 71 | 1.000 | S_{2,1}^{c} | | |
| johnston-linear-matrix-algebra_FO0849 | 71 | 1.000 | \left(S_{i, j}^{c}\right)^{n}=S_{i, j}^{n c} | | |
| johnston-linear-matrix-algebra_FO0850 | 71 | 1.000 | n \geq 1 | | |
| johnston-linear-matrix-algebra_FO0851 | 71 | 1.000 | S_{i, j}^{c}=I+c E_{i, j} | | |
| johnston-linear-matrix-algebra_FO0852 | 71 | 1.000 | E_{i, j} | | |
| johnston-linear-matrix-algebra_FO0853 | 71 | 1.000 | E_{i, j}^{2} | | |
| johnston-linear-matrix-algebra_FO0854 | 71 | 0.999 | R_{\mathbf{u}}^{\theta} | | |
| johnston-linear-matrix-algebra_FO0855 | 71 | 1.000 | \left[R_{\mathbf{u}}^{\theta}\right] | | |
| johnston-linear-matrix-algebra_FO0856 | 71 | 1.000 | \mathbf{u}=(1,0,0) | | |
| johnston-linear-matrix-algebra_FO0857 | 71 | 1.000 | R_{\mathbf{u}}^{\pi / 4}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0858 | 71 | 1.000 | \mathbf{u}=(1,2,2) / 3 | | |
| johnston-linear-matrix-algebra_FO0859 | 71 | 1.000 | \mathbf{v}=(3,2,1) | | |
| johnston-linear-matrix-algebra_FO0860 | 71 | 1.000 | R_{y z}^{\pi / 3} \circ R_{x y}^{\pi / 3} | | |
| johnston-linear-matrix-algebra_FO0861 | 71 | 1.000 | R_{\mathbf{u}}^{\theta}=R_{y z}^{\pi / 3} \circ R_{x y}^{\pi / 3} | | |
| johnston-linear-matrix-algebra_FO0862 | 73 | 1.000 | (A B)^{T}=A^{T} B^{T} | | |
| johnston-linear-matrix-algebra_FO0863 | 73 | 1.000 | (A B)^{3}=A^{3} B^{3} | | |
| johnston-linear-matrix-algebra_FO0864 | 73 | 1.000 | (A+B)(A-B)=A^{2}-B^{2} | | |
| johnston-linear-matrix-algebra_FO0865 | 73 | 0.993 | A^{2}=O | | |
| johnston-linear-matrix-algebra_FO0866 | 73 | 0.998 | R, S, T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO0867 | 73 | 1.000 | R \circ(S \circ T)=(R \circ S) \circ T | | |
| johnston-linear-matrix-algebra_FO0868 | 73 | 1.000 | \|A \mathbf{v}\|=\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0869 | 73 | 0.717 | \left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0870 | 73 | 1.000 | \left[\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0871 | 73 | 0.822 | \frac{1}{\sqrt{2}}\left[\begin{array}{cc}1 & -1 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0872 | 73 | 1.000 | \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0873 | 73 | 0.902 | \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0874 | 73 | 1.000 | \frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0875 | 73 | 0.931 | \frac{1}{9}\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 4 & 4 \\ 2 & 4 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0876 | 73 | 1.000 | \frac{1}{9}\left[\begin{array}{ccc}-1 & 4 & 8 \\ 4 & -7 & 4 \\ 8 & 4 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0877 | 73 | 1.000 | A^{500} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0878 | 73 | 1.000 | \mathbf{w}=\left(v_{2},-v_{1}\right) | | |
| johnston-linear-matrix-algebra_FO0879 | 73 | 1.000 | \mathbf{v} \cdot \mathbf{w}=v_{1} v_{2}-v_{2} v_{1}=0 | | |
| johnston-linear-matrix-algebra_FO0881 | 73 | 1.000 | \mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0882 | 73 | 1.000 | \mathbf{v} \times \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO0883 | 74 | 0.999 | \mathbf{w} \cdot(\mathbf{v} \times \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0884 | 74 | 0.717 | v_{1}, v_{2}, v_{3}, v_{1}, v_{2}, v_{3} | | |
| johnston-linear-matrix-algebra_FO0885 | 74 | 0.539 | 2 \times 6 | | |
| johnston-linear-matrix-algebra_FO0886 | 75 | 0.986 | (2,-1) | | |
| johnston-linear-matrix-algebra_FO0887 | 75 | 0.986 | (1,2) \cdot(2,-1)=2-2=0 | | |
| johnston-linear-matrix-algebra_FO0888 | 75 | 1.000 | (1,2) | | |
| johnston-linear-matrix-algebra_FO0889 | 75 | 0.999 | (1,2,0) \times(0,0,1)=(2,-1,0) | | |
| johnston-linear-matrix-algebra_FO0890 | 75 | 1.000 | (1,2,3) \times(-1,0,1)=(2,-4,2) | | |
| johnston-linear-matrix-algebra_FO0891 | 75 | 1.000 | (2,-4,2)=0 | | |
| johnston-linear-matrix-algebra_FO0892 | 75 | 1.000 | (-1,0,1) \cdot(2,-4,2)=0 | | |
| johnston-linear-matrix-algebra_FO0893 | 75 | 1.000 | \mathbf{v}, \mathbf{w}, \mathbf{x} \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0894 | 75 | 1.000 | \mathbf{v} \times \mathbf{w}=-(\mathbf{w} \times \mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO0895 | 75 | 1.000 | \mathbf{v} \times(\mathbf{w}+\mathbf{x})=\mathbf{v} \times \mathbf{w}+\mathbf{v} \times \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0896 | 75 | 1.000 | \mathbf{v} \times \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0897 | 75 | 1.000 | (c \mathbf{v}) \times \mathbf{w}=c(\mathbf{v} \times \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0898 | 75 | 1.000 | \mathbf{w} \times \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO0899 | 76 | 1.000 | v_{1}^{2} | | |
| johnston-linear-matrix-algebra_FO0900 | 76 | 1.000 | w_{1}^{2}, v_{2}^{2} w_{2}^{2} | | |
| johnston-linear-matrix-algebra_FO0901 | 76 | 1.000 | v_{3}^{2} w_{3}^{2} | | |
| johnston-linear-matrix-algebra_FO0902 | 76 | 1.000 | \|\mathbf{v} \times \mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0903 | 76 | 1.000 | \sqrt{\|\mathbf{v}\|^{2}\|\mathbf{w}\|^{2}-(\mathbf{v} \cdot \mathbf{w})^{2}} | | |
| johnston-linear-matrix-algebra_FO0904 | 76 | 0.934 | \|\mathbf{v}\|\|\mathbf{w}\| \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO0905 | 76 | 0.990 | \|\mathbf{w}\| \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO0906 | 76 | 1.000 | \|\mathbf{w}\| \cos (\theta) | | |
| johnston-linear-matrix-algebra_FO0907 | 76 | 1.000 | \|\mathbf{v} \times \mathbf{w}\|^{2}=\|\mathbf{v}\|^{2}\|\mathbf{w}\|^{2}-(\mathbf{v} \cdot \mathbf{w})^{2} | | |
| johnston-linear-matrix-algebra_FO0908 | 76 | 1.000 | \|\mathbf{v}\|^{2}\|\mathbf{w}\|^{2}-(\mathbf{v} \cdot \mathbf{w})^{2} | | |
| johnston-linear-matrix-algebra_FO0909 | 76 | 1.000 | \|\mathbf{v} \times \mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO0910 | 77 | 1.000 | \sin ^{2}(\theta)+\cos ^{2}(\theta)=1 | | |
| johnston-linear-matrix-algebra_FO0911 | 77 | 0.718 | 1-\cos ^{2}(\theta)=\sin ^{2}(\theta) | | |
| johnston-linear-matrix-algebra_FO0912 | 77 | 0.754 | \theta=\arccos (\mathbf{v} \cdot \mathbf{w} /(\|\mathbf{v}\|\|\mathbf{w}\|) | | |
| johnston-linear-matrix-algebra_FO0913 | 77 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| \cos (\theta) | | |
| johnston-linear-matrix-algebra_FO0914 | 77 | 1.000 | \sqrt{\sin ^{2}(\theta)}=|\sin (\theta)|=\sin (\theta) | | |
| johnston-linear-matrix-algebra_FO0915 | 77 | 1.000 | \sin (\theta) \geq 0 | | |
| johnston-linear-matrix-algebra_FO0916 | 77 | 0.883 | (3,1) | | |
| johnston-linear-matrix-algebra_FO0917 | 77 | 0.882 | \mathbb{R}^{4} | | |
| johnston-linear-matrix-algebra_FO0918 | 77 | 1.000 | (1,2,-1) \times(2,1,2)= | | |
| johnston-linear-matrix-algebra_FO0919 | 77 | 0.450 | \sqrt{5^{2}+(-4)^{2}+(-3)^{2}}=5 \sqrt{2} | | |
| johnston-linear-matrix-algebra_FO0920 | 77 | 1.000 | (1,2,0) \times | | |
| johnston-linear-matrix-algebra_FO0921 | 77 | 0.991 | (3,1,0)=(0,0,-5) | | |
| johnston-linear-matrix-algebra_FO0922 | 78 | 1.000 | |\cos (\theta)| | | |
| johnston-linear-matrix-algebra_FO0923 | 78 | 1.000 | \mathbf{w} \times \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0924 | 78 | 1.000 | \mathbf{v} \cdot(\mathbf{w} \times \mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0925 | 78 | 0.960 | |\mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})| | | |
| johnston-linear-matrix-algebra_FO0926 | 78 | 0.935 | |\mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})|=\|\mathbf{v}\|\|\mathbf{w} \times \mathbf{x}\||\cos (\theta)| | | |
| johnston-linear-matrix-algebra_FO0928 | 78 | 0.878 | \|\mathbf{v}\||\cos (\theta)| | | |
| johnston-linear-matrix-algebra_FO0929 | 79 | 0.998 | \mathbf{w}=(3,2,1) | | |
| johnston-linear-matrix-algebra_FO0930 | 79 | 1.000 | \mathbf{v}=(87,17,-43) | | |
| johnston-linear-matrix-algebra_FO0931 | 79 | 1.000 | \mathbf{w}=(87,17,-43) | | |
| johnston-linear-matrix-algebra_FO0932 | 79 | 1.000 | \mathbf{v}=(-1,2,0) | | |
| johnston-linear-matrix-algebra_FO0933 | 79 | 1.000 | \mathbf{w}=(-4,1,-2) | | |
| johnston-linear-matrix-algebra_FO0934 | 79 | 1.000 | \mathbf{v}=(1,0,0) | | |
| johnston-linear-matrix-algebra_FO0935 | 79 | 1.000 | \mathbf{w}=(0,1,0) | | |
| johnston-linear-matrix-algebra_FO0936 | 79 | 0.994 | \mathbf{v}=(2,1) | | |
| johnston-linear-matrix-algebra_FO0937 | 79 | 0.994 | \mathbf{w}=(-2,3) | | |
| johnston-linear-matrix-algebra_FO0938 | 79 | 1.000 | \mathbf{v}=(3,0,0) | | |
| johnston-linear-matrix-algebra_FO0939 | 79 | 1.000 | \mathbf{w}=(0,4,0) | | |
| johnston-linear-matrix-algebra_FO0940 | 79 | 1.000 | \mathbf{w}=(3,-1,2) | | |
| johnston-linear-matrix-algebra_FO0941 | 79 | 1.000 | \mathbf{v}=(1,0,1,-2) | | |
| johnston-linear-matrix-algebra_FO0942 | 79 | 1.000 | \mathbf{w}=(-2,1,3,1) | | |
| johnston-linear-matrix-algebra_FO0943 | 79 | 0.954 | \mathbf{v}=(0,4) | | |
| johnston-linear-matrix-algebra_FO0944 | 79 | 0.954 | \mathbf{w}=(1,1) | | |
| johnston-linear-matrix-algebra_FO0945 | 79 | 1.000 | \mathbf{v}=(0,2,2) | | |
| johnston-linear-matrix-algebra_FO0946 | 79 | 1.000 | \mathbf{w}=(1,-2,1) | | |
| johnston-linear-matrix-algebra_FO0947 | 79 | 0.650 | \mathbf{v}=(-1,1,-1) | | |
| johnston-linear-matrix-algebra_FO0948 | 79 | 1.000 | \mathbf{v}=(1,2,1,2) | | |
| johnston-linear-matrix-algebra_FO0949 | 79 | 1.000 | \mathbf{w}=(0,-1,2,1) | | |
| johnston-linear-matrix-algebra_FO0950 | 79 | 0.988 | \mathbf{v}=(1,0,0), \mathbf{w}=(0,2,0) | | |
| johnston-linear-matrix-algebra_FO0951 | 79 | 0.988 | \mathbf{x}=(0,0,3) | | |
| johnston-linear-matrix-algebra_FO0952 | 79 | 1.000 | \mathbf{v}=(0,4,1), \mathbf{w}=(1,1,0) | | |
| johnston-linear-matrix-algebra_FO0953 | 79 | 1.000 | \mathbf{x}=(2,0,-1) | | |
| johnston-linear-matrix-algebra_FO0954 | 79 | 0.913 | \mathbf{v}=(1,1,1), \mathbf{w}=(2,-1,1) | | |
| johnston-linear-matrix-algebra_FO0955 | 79 | 0.913 | \mathbf{x}=(2,-2,-1) | | |
| johnston-linear-matrix-algebra_FO0956 | 79 | 1.000 | \mathbf{v}=(-2,1,3), \mathbf{w}=(1,-4,2) | | |
| johnston-linear-matrix-algebra_FO0957 | 79 | 1.000 | \mathbf{x}=(3,3,-2) | | |
| johnston-linear-matrix-algebra_FO0958 | 79 | 1.000 | \left.\begin{array}{c}-2 \\ 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0959 | 79 | 0.455 | \begin{aligned} & {\left[\begin{array}{ll}1 & 7 \\ 0 & 1\end{array}\right]} \\ & \text { (b) }\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\end{aligned} | | |
| johnston-linear-matrix-algebra_FO0960 | 79 | 1.000 | \left[\begin{array}{ccc}1 & -3 & 2 \\ 0 & 2 & -7 \\ 0 & 0 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0961 | 79 | 1.000 | \left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO0962 | 79 | 1.000 | \|\mathbf{v} \times \mathbf{w}\|=\|\mathbf{w} \times \mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO0963 | 79 | 0.991 | \mathbf{v} \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO0964 | 79 | 0.991 | \mathbf{v} \times \mathbf{v}=\mathbf{v}^{2} | | |
| johnston-linear-matrix-algebra_FO0965 | 79 | 1.000 | (\mathbf{v}+\mathbf{v}) \times \mathbf{w}=2(\mathbf{v} \times \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0966 | 79 | 1.000 | \|\mathbf{v}\|=1,\|\mathbf{w}\|=1 | | |
| johnston-linear-matrix-algebra_FO0967 | 79 | 1.000 | \mathbf{v} \times \mathbf{w}=(1,1,1) | | |
| johnston-linear-matrix-algebra_FO0968 | 79 | 1.000 | \mathbf{e}_{1} \times \mathbf{e}_{2}, \mathbf{e}_{2} \times \mathbf{e}_{3} | | |
| johnston-linear-matrix-algebra_FO0969 | 79 | 1.000 | \mathbf{e}_{3} \times \mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO0970 | 79 | 1.000 | \mathbf{v} \times \mathbf{w}=(1 / 3,2 / 3,2 / 3) | | |
| johnston-linear-matrix-algebra_FO0971 | 80 | 0.999 | \|\mathbf{v} \times \mathbf{w}\|=\|\mathbf{v}\|\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO0972 | 80 | 0.560 | * * 1 . A .10 | | |
| johnston-linear-matrix-algebra_FO0973 | 80 | 0.560 | \mathbf{v} \cdot(\mathbf{v} \times \mathbf{w})=0 | | |
| johnston-linear-matrix-algebra_FO0974 | 80 | 1.000 | \mathbf{w} \cdot(\mathbf{v} \times \mathbf{w})=0 | | |
| johnston-linear-matrix-algebra_FO0975 | 80 | 1.000 | \mathbf{v} \times(c \mathbf{w})=c(\mathbf{v} \times \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0976 | 80 | 1.000 | (\mathbf{v} \times \mathbf{w}) \times \mathbf{x}=\mathbf{v} \times(\mathbf{w} \times \mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO0977 | 80 | 1.000 | \mathbf{v} \times \mathbf{w}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO0978 | 80 | 1.000 | \mathbf{v} \times \mathbf{w}=\mathbf{v} \times \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0979 | 80 | 1.000 | \mathbf{w} \neq \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0980 | 80 | 1.000 | \mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})=\mathbf{w} \cdot(\mathbf{x} \times \mathbf{v})=\mathbf{x} \cdot(\mathbf{v} \times \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO0981 | 80 | 1.000 | \mathbf{v} \times(\mathbf{w} \times \mathbf{x})=(\mathbf{v} \cdot \mathbf{x}) \mathbf{w}-(\mathbf{v} \cdot \mathbf{w}) \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO0982 | 81 | 1.000 | A-D-B | | |
| johnston-linear-matrix-algebra_FO0983 | 82 | 1.000 | a_{i, j}=a_{j, i} | | |
| johnston-linear-matrix-algebra_FO0984 | 82 | 1.000 | i, j | | |
| johnston-linear-matrix-algebra_FO0985 | 82 | 0.999 | \left.A=A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO0986 | 82 | 1.000 | a_{i, j}=1 | | |
| johnston-linear-matrix-algebra_FO0987 | 82 | 1.000 | a_{i, j}=0 | | |
| johnston-linear-matrix-algebra_FO0988 | 83 | 1.000 | \left[A^{k}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO0989 | 83 | 1.000 | k \geq 1 | | |
| johnston-linear-matrix-algebra_FO0990 | 83 | 1.000 | k=1 | | |
| johnston-linear-matrix-algebra_FO0991 | 83 | 1.000 | k=2 | | |
| johnston-linear-matrix-algebra_FO0992 | 83 | 0.981 | a_{i, 1} a_{1, j} | | |
| johnston-linear-matrix-algebra_FO0993 | 83 | 1.000 | a_{i, 2} a_{2, j} | | |
| johnston-linear-matrix-algebra_FO0994 | 83 | 1.000 | \left[A^{2}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO0995 | 84 | 1.000 | A^{6} | | |
| johnston-linear-matrix-algebra_FO0996 | 84 | 0.968 | (A) | | |
| johnston-linear-matrix-algebra_FO0997 | 84 | 0.968 | (D) | | |
| johnston-linear-matrix-algebra_FO0998 | 84 | 1.000 | A^{6}=A A A A A A | | |
| johnston-linear-matrix-algebra_FO0999 | 84 | 1.000 | A^{6}=\left(A^{3}\right)^{2} | | |
| johnston-linear-matrix-algebra_FO1000 | 84 | 1.000 | (B) | | |
| johnston-linear-matrix-algebra_FO1001 | 84 | 1.000 | (C) | | |
| johnston-linear-matrix-algebra_FO1002 | 85 | 1.000 | \left[A^{2}\right]_{5,1} | | |
| johnston-linear-matrix-algebra_FO1003 | 85 | 1.000 | \left[A^{3}\right]_{4,6}+\left[A^{2}\right]_{4,6}+a_{4,6} | | |
| johnston-linear-matrix-algebra_FO1004 | 85 | 1.000 | \left[A^{3}\right]_{4,6} | | |
| johnston-linear-matrix-algebra_FO1005 | 86 | 1.000 | E | | |
| johnston-linear-matrix-algebra_FO1006 | 86 | 1.000 | \left[A^{4}\right]_{3,5} | | |
| johnston-linear-matrix-algebra_FO1007 | 86 | 0.995 | A^{87} | | |
| johnston-linear-matrix-algebra_FO1008 | 87 | 0.995 | 0,0,1,0,0 | | |
| johnston-linear-matrix-algebra_FO1009 | 87 | 1.000 | \left[A^{2}\right]_{1,4} | | |
| johnston-linear-matrix-algebra_FO1010 | 88 | 1.000 | \left[A^{3}\right]_{2,3} | | |
| johnston-linear-matrix-algebra_FO1011 | 88 | 1.000 | (0,1,1,1) \cdot(1,1,2,1)=4 | | |
| johnston-linear-matrix-algebra_FO1012 | 88 | 1.000 | \sum_{j=1}^{n} w_{j} | | |
| johnston-linear-matrix-algebra_FO1013 | 88 | 1.000 | w_{1}+w_{2}+\cdots+w_{n} | | |
| johnston-linear-matrix-algebra_FO1014 | 88 | 0.964 | n \approx 7.5 | | |
| johnston-linear-matrix-algebra_FO1015 | 88 | 1.000 | 1 \leq j \leq n, w_{j} | | |
| johnston-linear-matrix-algebra_FO1016 | 89 | 1.000 | (3+2+1+2) / 4=2 | | |
| johnston-linear-matrix-algebra_FO1017 | 89 | 1.000 | 2 \leq 2.25 | | |
| johnston-linear-matrix-algebra_FO1018 | 89 | 0.999 | 2 \times 2=4 | | |
| johnston-linear-matrix-algebra_FO1019 | 89 | 0.835 | 1 \times 1=1 | | |
| johnston-linear-matrix-algebra_FO1020 | 89 | 1.000 | (2+1+2+3+2+3+3+2) / 8= | | |
| johnston-linear-matrix-algebra_FO1021 | 89 | 1.000 | w_{j} | | |
| johnston-linear-matrix-algebra_FO1022 | 89 | 1.000 | w_{j}^{2} | | |
| johnston-linear-matrix-algebra_FO1023 | 89 | 1.000 | \sum_{j=1}^{n} w_{j}^{2} | | |
| johnston-linear-matrix-algebra_FO1024 | 89 | 1.000 | \mathbf{v}=(1,1, \ldots, 1) \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1025 | 89 | 1.000 | n \sum_{j=1}^{n} w_{j} | | |
| johnston-linear-matrix-algebra_FO1026 | 90 | 1.000 | A=A^{T} | | |
| johnston-linear-matrix-algebra_FO1027 | 90 | 1.000 | \left[\begin{array}{llll}0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1028 | 90 | 1.000 | \left[\begin{array}{lllll}0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1029 | 90 | 1.000 | \left[\begin{array}{llll}2 & 0 & 0 & 1 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 1 \\ 3 & 0 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1030 | 91 | 1.000 | A^{n} | | |
| johnston-linear-matrix-algebra_FO1031 | 92 | 1.000 | x, y | | |
| johnston-linear-matrix-algebra_FO1032 | 92 | 1.000 | x_{1}, x_{2}, \ldots, x_{n} | | |
| johnston-linear-matrix-algebra_FO1033 | 92 | 1.000 | a_{1}, a_{2}, \ldots, a_{n} | | |
| johnston-linear-matrix-algebra_FO1034 | 92 | 1.000 | 4 x-6 y=-3 | | |
| johnston-linear-matrix-algebra_FO1035 | 93 | 1.000 | y=m x+b | | |
| johnston-linear-matrix-algebra_FO1036 | 93 | 1.000 | x=2 | | |
| johnston-linear-matrix-algebra_FO1037 | 93 | 1.000 | y=1 | | |
| johnston-linear-matrix-algebra_FO1038 | 93 | 1.000 | a x+b y=c | | |
| johnston-linear-matrix-algebra_FO1039 | 93 | 1.000 | a x+b y+c z=d | | |
| johnston-linear-matrix-algebra_FO1040 | 93 | 1.000 | \mathbf{x}= | | |
| johnston-linear-matrix-algebra_FO1041 | 93 | 1.000 | \left(x_{1}, x_{2}, \ldots, x_{n}\right) | | |
| johnston-linear-matrix-algebra_FO1042 | 94 | 0.834 | \mathbf{x}=(2,1) | | |
| johnston-linear-matrix-algebra_FO1043 | 94 | 1.000 | 3 y=3 | | |
| johnston-linear-matrix-algebra_FO1044 | 94 | 1.000 | x+2 y=4 | | |
| johnston-linear-matrix-algebra_FO1046 | 94 | 0.943 | x+2 y | | |
| johnston-linear-matrix-algebra_FO1047 | 95 | 1.000 | A \mathbf{x}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1048 | 95 | 0.950 | a_{i, j}, \mathbf{b}=\left(b_{1}, b_{2}, \ldots, b_{m}\right) \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO1049 | 95 | 0.997 | \mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1050 | 96 | 0.743 | \mathbf{x}_{1} | | |
| johnston-linear-matrix-algebra_FO1051 | 96 | 0.743 | \mathbf{x}_{2} | | |
| johnston-linear-matrix-algebra_FO1052 | 96 | 1.000 | A \in \mathcal{M}_{m, n}, \mathbf{x} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1053 | 96 | 1.000 | \mathbf{x}_{1} \neq \mathbf{x}_{2} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1054 | 96 | 1.000 | A \mathbf{x}_{1}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1055 | 96 | 1.000 | A \mathbf{x}_{2}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1056 | 96 | 1.000 | (1-c) \mathbf{x}_{1}+c \mathbf{x}_{2} | | |
| johnston-linear-matrix-algebra_FO1057 | 96 | 1.000 | 3 z=6 | | |
| johnston-linear-matrix-algebra_FO1058 | 96 | 1.000 | z=2 | | |
| johnston-linear-matrix-algebra_FO1059 | 97 | 1.000 | 2 y=16 | | |
| johnston-linear-matrix-algebra_FO1060 | 97 | 1.000 | y=8 | | |
| johnston-linear-matrix-algebra_FO1061 | 97 | 1.000 | x+24=9 | | |
| johnston-linear-matrix-algebra_FO1062 | 97 | 1.000 | x=-15 | | |
| johnston-linear-matrix-algebra_FO1063 | 97 | 1.000 | (x, y, z)=(-15,8,2) | | |
| johnston-linear-matrix-algebra_FO1064 | 98 | 1.000 | y-3 z=2 | | |
| johnston-linear-matrix-algebra_FO1065 | 98 | 1.000 | [A \mid \mathbf{b}] | | |
| johnston-linear-matrix-algebra_FO1066 | 99 | 1.000 | c R_{j} | | |
| johnston-linear-matrix-algebra_FO1067 | 99 | 1.000 | R_{i} \leftrightarrow R_{j} | | |
| johnston-linear-matrix-algebra_FO1068 | 99 | 0.974 | )+c( | | |
| johnston-linear-matrix-algebra_FO1069 | 99 | 0.974 | ) | | |
| johnston-linear-matrix-algebra_FO1070 | 99 | 0.996 | R_{i}+c R_{j} | | |
| johnston-linear-matrix-algebra_FO1071 | 100 | 0.997 | (x, y, z)= | | |
| johnston-linear-matrix-algebra_FO1072 | 100 | 1.000 | \frac{1}{c} R_{j} | | |
| johnston-linear-matrix-algebra_FO1073 | 100 | 1.000 | R_{i}-c R_{j} | | |
| johnston-linear-matrix-algebra_FO1074 | 100 | 1.000 | c \neq 0 | | |
| johnston-linear-matrix-algebra_FO1075 | 100 | 1.000 | \mathbf{x}=(1,1) | | |
| johnston-linear-matrix-algebra_FO1076 | 100 | 1.000 | \mathbf{x}=(3,0) | | |
| johnston-linear-matrix-algebra_FO1077 | 101 | 1.000 | \left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1078 | 101 | 1.000 | \left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1079 | 101 | 1.000 | \left[\begin{array}{cccc}2 & 0 & -1 & 5 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1080 | 101 | 1.000 | \left[\begin{array}{cccc}1 & 2 & 3 & 4 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1081 | 104 | 1.000 | 0=1 | | |
| johnston-linear-matrix-algebra_FO1082 | 104 | 0.513 | \left[\left.\begin{array}{llll}0 & 0 & \cdots & 0\end{array} \right\rvert\, b\right] | | |
| johnston-linear-matrix-algebra_FO1083 | 104 | 1.000 | b \neq 0 | | |
| johnston-linear-matrix-algebra_FO1084 | 105 | 1.000 | x=3-4 y | | |
| johnston-linear-matrix-algebra_FO1085 | 105 | 1.000 | y=y | | |
| johnston-linear-matrix-algebra_FO1087 | 107 | 1.000 | \left[\begin{array}{cccc|c}1 & -1 & 0 & 1 & 2 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1088 | 107 | 0.997 | \left[\begin{array}{ccc|c}1 & 2 & -4 & -4 \\ 0 & 3 & -1 & 2 \\ 0 & 0 & 8 & 8\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1089 | 107 | 1.000 | w, x, y, z | | |
| johnston-linear-matrix-algebra_FO1090 | 107 | 1.000 | w | | |
| johnston-linear-matrix-algebra_FO1091 | 107 | 1.000 | w=2+x-z | | |
| johnston-linear-matrix-algebra_FO1092 | 107 | 1.000 | y=1+z | | |
| johnston-linear-matrix-algebra_FO1093 | 107 | 0.818 | \left[\left.\begin{array}{lll}0 & 0 & 0\end{array} \right\rvert\, b\right] | | |
| johnston-linear-matrix-algebra_FO1094 | 107 | 1.000 | x, y, z | | |
| johnston-linear-matrix-algebra_FO1095 | 107 | 1.000 | 8 z=8 | | |
| johnston-linear-matrix-algebra_FO1096 | 107 | 1.000 | z=1 | | |
| johnston-linear-matrix-algebra_FO1097 | 107 | 1.000 | 3 y-1=2 | | |
| johnston-linear-matrix-algebra_FO1098 | 107 | 1.000 | x+2-4=-4 | | |
| johnston-linear-matrix-algebra_FO1099 | 107 | 1.000 | x=-2 | | |
| johnston-linear-matrix-algebra_FO1100 | 107 | 1.000 | (x, y, z)=(-2,1,1) | | |
| johnston-linear-matrix-algebra_FO1101 | 107 | 1.000 | 0 x+0 y+0 z=-42 / 5 | | |
| johnston-linear-matrix-algebra_FO1102 | 107 | 1.000 | A \mathbf{x}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1103 | 108 | 1.000 | m<n | | |
| johnston-linear-matrix-algebra_FO1104 | 108 | 1.000 | v_{1}=-2 v_{2}-3 v_{3} | | |
| johnston-linear-matrix-algebra_FO1105 | 108 | 1.000 | v_{2}=v_{3} | | |
| johnston-linear-matrix-algebra_FO1106 | 108 | 1.000 | \mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO1107 | 109 | 1.000 | v_{4} | | |
| johnston-linear-matrix-algebra_FO1108 | 109 | 1.000 | \left(v_{1}, v_{2}, v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO1109 | 109 | 1.000 | v_{1}=-2 v_{2}-3 v_{3}=-5 v_{3} | | |
| johnston-linear-matrix-algebra_FO1110 | 109 | 1.000 | v_{3}=1 | | |
| johnston-linear-matrix-algebra_FO1111 | 109 | 1.000 | \mathbf{v}=(-5,1,1) | | |
| johnston-linear-matrix-algebra_FO1112 | 109 | 0.999 | v_{1}, v_{2}, v_{3} | | |
| johnston-linear-matrix-algebra_FO1113 | 109 | 1.000 | \left(v_{1}, v_{2}, v_{3}, v_{4}\right) | | |
| johnston-linear-matrix-algebra_FO1114 | 109 | 1.000 | v_{1}= | | |
| johnston-linear-matrix-algebra_FO1115 | 109 | 0.991 | 0, v_{2}=-v_{4} / 2 | | |
| johnston-linear-matrix-algebra_FO1116 | 109 | 0.991 | v_{3}=-v_{4} / 2 | | |
| johnston-linear-matrix-algebra_FO1117 | 109 | 1.000 | v_{4}=2 | | |
| johnston-linear-matrix-algebra_FO1118 | 109 | 1.000 | \mathbf{v}=(0,-1,-1,2) | | |
| johnston-linear-matrix-algebra_FO1119 | 109 | 1.000 | c_{1}, c_{2} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1120 | 110 | 1.000 | c_{1}, c_{2} | | |
| johnston-linear-matrix-algebra_FO1121 | 110 | 1.000 | (1,3,1)=\frac{4}{5}(-1,3,2)+ | | |
| johnston-linear-matrix-algebra_FO1122 | 110 | 1.000 | \frac{3}{5}(3,1,-1) | | |
| johnston-linear-matrix-algebra_FO1123 | 110 | 1.000 | c_{1} | | |
| johnston-linear-matrix-algebra_FO1124 | 110 | 1.000 | c_{2} | | |
| johnston-linear-matrix-algebra_FO1125 | 110 | 1.000 | 0=3 | | |
| johnston-linear-matrix-algebra_FO1126 | 110 | 1.000 | c_{2}=3 / 5 | | |
| johnston-linear-matrix-algebra_FO1127 | 110 | 1.000 | -c_{1}+3 c_{2}=1 | | |
| johnston-linear-matrix-algebra_FO1128 | 110 | 1.000 | c_{1}=4 / 5 | | |
| johnston-linear-matrix-algebra_FO1129 | 110 | 1.000 | 0.02 \times 500=10 | | |
| johnston-linear-matrix-algebra_FO1130 | 111 | 1.000 | x=200 | | |
| johnston-linear-matrix-algebra_FO1131 | 111 | 0.996 | 3.5 \% | | |
| johnston-linear-matrix-algebra_FO1132 | 111 | 0.996 | y=300 | | |
| johnston-linear-matrix-algebra_FO1133 | 111 | 0.996 | 1 \% | | |
| johnston-linear-matrix-algebra_FO1134 | 111 | 1.000 | (x, y)=(200,300) | | |
| johnston-linear-matrix-algebra_FO1135 | 111 | 1.000 | (x, y)=(125,375) | | |
| johnston-linear-matrix-algebra_FO1136 | 112 | 1.000 | \mathrm{H}_{2} \mathrm{O} | | |
| johnston-linear-matrix-algebra_FO1137 | 112 | 0.996 | \left(\mathrm{C}_{4} \mathrm{H}_{10}\right) | | |
| johnston-linear-matrix-algebra_FO1138 | 112 | 0.996 | \left(\mathrm{O}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO1139 | 112 | 0.890 | \left(\mathrm{CO}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO1140 | 112 | 0.890 | \left(\mathrm{H}_{2} \mathrm{O}\right) | | |
| johnston-linear-matrix-algebra_FO1141 | 112 | 1.000 | w, x, y | | |
| johnston-linear-matrix-algebra_FO1142 | 112 | 1.000 | 2 x | | |
| johnston-linear-matrix-algebra_FO1143 | 112 | 1.000 | 2 y+z | | |
| johnston-linear-matrix-algebra_FO1144 | 112 | 1.000 | z=10 | | |
| johnston-linear-matrix-algebra_FO1145 | 112 | 1.000 | w=2, x=13 | | |
| johnston-linear-matrix-algebra_FO1146 | 113 | 1.000 | \left[\begin{array}{ccc}4 & 0 & 8 \\ 4 & 1 & 11 \\ 1 & 2 & 8 \\ -1 & 0 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1147 | 113 | 1.000 | 4 x-\sin (1) y+2 z=\sqrt[3]{5} | | |
| johnston-linear-matrix-algebra_FO1148 | 113 | 1.000 | x y+4 z=4 | | |
| johnston-linear-matrix-algebra_FO1149 | 114 | 1.000 | t \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1150 | 114 | 1.000 | t | | |
| johnston-linear-matrix-algebra_FO1151 | 114 | 1.000 | h, k \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1152 | 114 | 1.000 | h | | |
| johnston-linear-matrix-algebra_FO1153 | 114 | 1.000 | h=0.95, h=1.00 | | |
| johnston-linear-matrix-algebra_FO1154 | 114 | 1.000 | h=1.05 | | |
| johnston-linear-matrix-algebra_FO1155 | 114 | 1.000 | a, b, c, d \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1156 | 114 | 1.000 | a d-b c \neq 0 | | |
| johnston-linear-matrix-algebra_FO1157 | 114 | 1.000 | A, B, R \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO1158 | 114 | 1.000 | R_{2}-R_{1} | | |
| johnston-linear-matrix-algebra_FO1159 | 114 | 1.000 | R_{1}-R_{2} | | |
| johnston-linear-matrix-algebra_FO1160 | 114 | 0.998 | \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1161 | 114 | 0.999 | \mathbf{b}=(3,4), \mathbf{v}_{1}=(6,8) | | |
| johnston-linear-matrix-algebra_FO1162 | 114 | 1.000 | \mathbf{b}=(2,-3), \mathbf{v}_{1}=(1,2), \mathbf{v}_{2}=(4,-1) | | |
| johnston-linear-matrix-algebra_FO1163 | 114 | 1.000 | \mathbf{b}=(1,5,-6), \mathbf{v}_{1}=(1,0,0), \mathbf{v}_{2}=(0,1,0), \mathbf{v}_{3}= | | |
| johnston-linear-matrix-algebra_FO1164 | 114 | 1.000 | \mathbf{b}=(2,1,2), \mathbf{v}_{1}=(-2,2,1), \mathbf{v}_{2}=(1,2,3) | | |
| johnston-linear-matrix-algebra_FO1165 | 114 | 1.000 | \mathbf{b}=(2,1,2), \mathbf{v}_{1}=(-4,4,-1), \mathbf{v}_{2}=(2,-1,1) | | |
| johnston-linear-matrix-algebra_FO1166 | 114 | 1.000 | \mathbf{b}=(2,1,2,3), \mathbf{v}_{1}=(1,2,3,4), \mathbf{v}_{2}=(4,3,2,1) | | |
| johnston-linear-matrix-algebra_FO1167 | 114 | 1.000 | \mathbf{v}_{3}=(1,-1,1,-1) | | |
| johnston-linear-matrix-algebra_FO1168 | 114 | 0.831 | \mathbf{b}=(1,3,-3,-1), \mathbf{v}_{1}=(1,2,3,4), \mathbf{v}_{2}=(4,3,2,1) | | |
| johnston-linear-matrix-algebra_FO1169 | 114 | 1.000 | \mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right) | | |
| johnston-linear-matrix-algebra_FO1170 | 115 | 1.000 | \left(\mathrm{SO}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO1171 | 115 | 0.996 | \left(\mathrm{C}_{2} \mathrm{H}_{6}\right) | | |
| johnston-linear-matrix-algebra_FO1172 | 115 | 1.000 | P_{\mathbf{u}}(\mathbf{v})=(2,4,4) | | |
| johnston-linear-matrix-algebra_FO1173 | 115 | 1.000 | (3,-2,1)=c_{1}(1,4,2)+ | | |
| johnston-linear-matrix-algebra_FO1174 | 115 | 1.000 | c_{2}(2,-1,1) | | |
| johnston-linear-matrix-algebra_FO1175 | 115 | 0.566 | T(1,1)=(3,7), T(1,-1)=(-1,-1) | | |
| johnston-linear-matrix-algebra_FO1176 | 115 | 1.000 | T(1,2)=(5,3,4), T(2,-1)=(0,1,3) | | |
| johnston-linear-matrix-algebra_FO1177 | 115 | 0.992 | T(1,1,1)=(4,6,1), \quad T(2,-1,1)=(1,1,-4) | | |
| johnston-linear-matrix-algebra_FO1178 | 115 | 1.000 | T(0,0,1)=(1,2,0) | | |
| johnston-linear-matrix-algebra_FO1179 | 115 | 1.000 | A B=B A | | |
| johnston-linear-matrix-algebra_FO1180 | 115 | 1.000 | B \in \mathcal{M}_{2} | | |
| johnston-linear-matrix-algebra_FO1181 | 115 | 1.000 | D \in \mathcal{M}_{2} | | |
| johnston-linear-matrix-algebra_FO1182 | 115 | 1.000 | C D=D C | | |
| johnston-linear-matrix-algebra_FO1183 | 115 | 1.000 | c I | | |
| johnston-linear-matrix-algebra_FO1184 | 115 | 1.000 | 2=m+b | | |
| johnston-linear-matrix-algebra_FO1185 | 115 | 1.000 | (3,8) | | |
| johnston-linear-matrix-algebra_FO1186 | 115 | 1.000 | 8=3 m+b | | |
| johnston-linear-matrix-algebra_FO1187 | 115 | 0.977 | y=a x^{2}+b x+c | | |
| johnston-linear-matrix-algebra_FO1188 | 115 | 1.000 | (2,6) | | |
| johnston-linear-matrix-algebra_FO1189 | 115 | 1.000 | 6=4 a+2 b+c | | |
| johnston-linear-matrix-algebra_FO1190 | 115 | 1.000 | y=a x^{3}+b x^{2}+c x+d | | |
| johnston-linear-matrix-algebra_FO1191 | 116 | 1.000 | a, b, c, d | | |
| johnston-linear-matrix-algebra_FO1192 | 116 | 1.000 | e | | |
| johnston-linear-matrix-algebra_FO1193 | 116 | 1.000 | p(0)=3.70, p(1)=4.46 | | |
| johnston-linear-matrix-algebra_FO1194 | 117 | 1.000 | R_{1} \leftrightarrow R_{2} | | |
| johnston-linear-matrix-algebra_FO1195 | 117 | 1.000 | R_{3}-3 R_{1} | | |
| johnston-linear-matrix-algebra_FO1196 | 117 | 1.000 | \frac{1}{2} R_{2} | | |
| johnston-linear-matrix-algebra_FO1197 | 117 | 1.000 | R_{i} \leftrightarrow | | |
| johnston-linear-matrix-algebra_FO1198 | 117 | 1.000 | R_{j} | | |
| johnston-linear-matrix-algebra_FO1200 | 119 | 0.512 | E A=R | | |
| johnston-linear-matrix-algebra_FO1201 | 119 | 1.000 | R_{3}-R_{2} | | |
| johnston-linear-matrix-algebra_FO1202 | 119 | 1.000 | \frac{1}{4} R_{3} | | |
| johnston-linear-matrix-algebra_FO1203 | 119 | 1.000 | R_{1}+R_{3} | | |
| johnston-linear-matrix-algebra_FO1204 | 119 | 1.000 | E=E_{7} E_{6} E_{5} E_{4} E_{3} E_{2} E_{1} | | |
| johnston-linear-matrix-algebra_FO1205 | 120 | 0.999 | E_{1}, E_{2}, \ldots, E_{7} | | |
| johnston-linear-matrix-algebra_FO1206 | 120 | 1.000 | E_{1} | | |
| johnston-linear-matrix-algebra_FO1207 | 120 | 1.000 | E_{2} | | |
| johnston-linear-matrix-algebra_FO1208 | 120 | 0.516 | -5 / 8 | | |
| johnston-linear-matrix-algebra_FO1209 | 120 | 0.516 | 2 / 8=1 / 4 | | |
| johnston-linear-matrix-algebra_FO1210 | 120 | 0.516 | 1 / 4 | | |
| johnston-linear-matrix-algebra_FO1211 | 120 | 1.000 | E= | | |
| johnston-linear-matrix-algebra_FO1212 | 120 | 0.984 | E_{7} E_{6} E_{5} E_{4} E_{3} E_{2} E_{1} | | |
| johnston-linear-matrix-algebra_FO1213 | 120 | 0.913 | [A \mid I] | | |
| johnston-linear-matrix-algebra_FO1214 | 121 | 1.000 | [R \mid E] | | |
| johnston-linear-matrix-algebra_FO1215 | 121 | 1.000 | A, R \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO1216 | 121 | 1.000 | E \in \mathcal{M}_{m} | | |
| johnston-linear-matrix-algebra_FO1217 | 121 | 1.000 | R=E A | | |
| johnston-linear-matrix-algebra_FO1218 | 121 | 0.706 | E_{1}, E_{2}, \ldots, E_{k} | | |
| johnston-linear-matrix-algebra_FO1219 | 121 | 1.000 | E=E_{k} \cdots E_{2} E_{1} | | |
| johnston-linear-matrix-algebra_FO1220 | 121 | 1.000 | E_{1} E_{2}=I | | |
| johnston-linear-matrix-algebra_FO1221 | 121 | 1.000 | E_{2} E_{1}=I | | |
| johnston-linear-matrix-algebra_FO1222 | 122 | 1.000 | R_{3}+3 R_{1} | | |
| johnston-linear-matrix-algebra_FO1223 | 122 | 1.000 | A^{-1} | | |
| johnston-linear-matrix-algebra_FO1224 | 122 | 1.000 | B, C \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO1225 | 122 | 1.000 | A B=B A=I | | |
| johnston-linear-matrix-algebra_FO1226 | 122 | 1.000 | A C=C A=I | | |
| johnston-linear-matrix-algebra_FO1227 | 122 | 1.000 | \frac{1}{2}\left[\begin{array}{cc}-4 & 2 \\ 3 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1228 | 122 | 1.000 | \left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1229 | 122 | 1.000 | \left[\begin{array}{cc}-4 & 2 \\ 2 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1230 | 122 | 0.991 | \left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1231 | 122 | 0.991 | \frac{1}{2}\left[\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1232 | 123 | 1.000 | T^{-1} | | |
| johnston-linear-matrix-algebra_FO1233 | 123 | 1.000 | T \circ T^{-1}=T^{-1} \circ T=I | | |
| johnston-linear-matrix-algebra_FO1237 | 124 | 0.966 | 0 x | | |
| johnston-linear-matrix-algebra_FO1238 | 124 | 1.000 | \left(A^{-1}\right)^{-1}=A | | |
| johnston-linear-matrix-algebra_FO1239 | 124 | 1.000 | (c A)^{-1}=\frac{1}{c} A^{-1} | | |
| johnston-linear-matrix-algebra_FO1240 | 124 | 1.000 | \left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T} | | |
| johnston-linear-matrix-algebra_FO1241 | 124 | 1.000 | \left(A^{k}\right)^{-1}=\left(A^{-1}\right)^{k} | | |
| johnston-linear-matrix-algebra_FO1242 | 124 | 1.000 | \frac{1}{c} A^{-1} | | |
| johnston-linear-matrix-algebra_FO1243 | 124 | 1.000 | A^{k \ell}= | | |
| johnston-linear-matrix-algebra_FO1244 | 124 | 1.000 | \left(A^{k}\right)^{\ell} | | |
| johnston-linear-matrix-algebra_FO1245 | 124 | 0.965 | A^{-2} | | |
| johnston-linear-matrix-algebra_FO1246 | 125 | 1.000 | c R_{i} | | |
| johnston-linear-matrix-algebra_FO1247 | 125 | 1.000 | \left[\begin{array}{ll}1 & 0 \\ 0 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1248 | 125 | 1.000 | \left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1249 | 125 | 1.000 | \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1250 | 125 | 1.000 | \left[\begin{array}{ccc}1 & 0 & 0 \\ -5 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1251 | 125 | 1.000 | 3 R_{2} | | |
| johnston-linear-matrix-algebra_FO1252 | 125 | 1.000 | \frac{1}{3} R_{2} | | |
| johnston-linear-matrix-algebra_FO1253 | 125 | 1.000 | R_{1}+2 R_{2} | | |
| johnston-linear-matrix-algebra_FO1254 | 125 | 1.000 | R_{1}-2 R_{2} | | |
| johnston-linear-matrix-algebra_FO1255 | 125 | 1.000 | R_{2} \leftrightarrow R_{3} | | |
| johnston-linear-matrix-algebra_FO1256 | 125 | 1.000 | R_{2}-5 R_{1} | | |
| johnston-linear-matrix-algebra_FO1257 | 126 | 1.000 | R_{2}+5 R_{1} | | |
| johnston-linear-matrix-algebra_FO1258 | 126 | 1.000 | B^{-1} A^{-1} | | |
| johnston-linear-matrix-algebra_FO1259 | 126 | 1.000 | \left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 0 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 6 \\ 0 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1260 | 127 | 1.000 | \mathbf{b} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1261 | 127 | 1.000 | \mathbf{x}=A^{-1} \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1262 | 127 | 1.000 | \mathbf{x}, \mathbf{y} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1263 | 127 | 1.000 | A \mathbf{y}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1264 | 127 | 1.000 | A(\mathbf{x}-\mathbf{y})=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1265 | 127 | 1.000 | \mathbf{x}=\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO1266 | 127 | 0.842 | \mathbf{b}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1267 | 127 | 1.000 | [A \mid \mathbf{0}] | | |
| johnston-linear-matrix-algebra_FO1268 | 127 | 1.000 | [R \mid \mathbf{0}] | | |
| johnston-linear-matrix-algebra_FO1269 | 127 | 1.000 | R \neq I | | |
| johnston-linear-matrix-algebra_FO1270 | 128 | 0.999 | \left[R \mid \mathbf{e}_{n}\right] | | |
| johnston-linear-matrix-algebra_FO1271 | 128 | 1.000 | \mathbf{e}_{n} | | |
| johnston-linear-matrix-algebra_FO1272 | 128 | 1.000 | R=I | | |
| johnston-linear-matrix-algebra_FO1273 | 128 | 0.996 | E_{k} \cdots E_{2} E_{1} A=I | | |
| johnston-linear-matrix-algebra_FO1274 | 128 | 1.000 | E_{k}^{-1}, E_{k-1}^{-1} | | |
| johnston-linear-matrix-algebra_FO1275 | 128 | 1.000 | E_{1}^{-1} | | |
| johnston-linear-matrix-algebra_FO1276 | 128 | 1.000 | \left[\begin{array}{ccc}1 & 2 & -2 \\ 0 & 1 & -2 \\ 1 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1277 | 129 | 1.000 | A^{-1}[A \mid I]=\left[I \mid A^{-1}\right] | | |
| johnston-linear-matrix-algebra_FO1278 | 129 | 1.000 | \left[I \mid A^{-1}\right] | | |
| johnston-linear-matrix-algebra_FO1279 | 129 | 1.000 | E \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO1280 | 129 | 1.000 | [I \mid E] | | |
| johnston-linear-matrix-algebra_FO1281 | 129 | 1.000 | A^{-1}=E | | |
| johnston-linear-matrix-algebra_FO1282 | 129 | 1.000 | I=E A | | |
| johnston-linear-matrix-algebra_FO1283 | 129 | 1.000 | E=A^{-1} | | |
| johnston-linear-matrix-algebra_FO1284 | 129 | 0.995 | \left[\begin{array}{ll}2 & 2 \\ 4 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1285 | 129 | 1.000 | \left[\begin{array}{cc}1 & -2 \\ -3 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1286 | 129 | 1.000 | \left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1287 | 131 | 0.500 | \mathbf{2} \times \mathbf{2} | | |
| johnston-linear-matrix-algebra_FO1288 | 131 | 1.000 | a d-b c | | |
| johnston-linear-matrix-algebra_FO1289 | 131 | 0.924 | A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1290 | 131 | 1.000 | a d-b c=0 | | |
| johnston-linear-matrix-algebra_FO1291 | 131 | 1.000 | a d=b c | | |
| johnston-linear-matrix-algebra_FO1292 | 131 | 1.000 | a=0 | | |
| johnston-linear-matrix-algebra_FO1293 | 131 | 1.000 | b=0 | | |
| johnston-linear-matrix-algebra_FO1294 | 131 | 1.000 | a, b \neq 0 | | |
| johnston-linear-matrix-algebra_FO1295 | 131 | 1.000 | d / b=c / a | | |
| johnston-linear-matrix-algebra_FO1296 | 131 | 1.000 | a d-b c=1 \times 4-2 \times 3=4-6=-2 \neq 0 | | |
| johnston-linear-matrix-algebra_FO1297 | 131 | 1.000 | a d-b c=1 \times 4-2 \times 2=4-4=0 | | |
| johnston-linear-matrix-algebra_FO1298 | 131 | 1.000 | a d-b c=2 \times 5-2 \times 4=10-8=2 \neq 0 | | |
| johnston-linear-matrix-algebra_FO1299 | 132 | 1.000 | a d-b c=1 \times 6-(-2) \times(-3)=6-6=0 | | |
| johnston-linear-matrix-algebra_FO1300 | 133 | 1.000 | A A^{-1}=A^{-1} A=I | | |
| johnston-linear-matrix-algebra_FO1301 | 133 | 1.000 | B \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO1302 | 133 | 1.000 | A B=I | | |
| johnston-linear-matrix-algebra_FO1303 | 133 | 1.000 | B A=I | | |
| johnston-linear-matrix-algebra_FO1304 | 133 | 1.000 | A^{-1}=B | | |
| johnston-linear-matrix-algebra_FO1305 | 133 | 0.974 | B A \mathbf{x}=B \mathbf{0}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1306 | 133 | 0.974 | B A \mathbf{x}=I \mathbf{x}=\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO1307 | 133 | 1.000 | B=A^{-1} | | |
| johnston-linear-matrix-algebra_FO1308 | 133 | 0.966 | A C=I | | |
| johnston-linear-matrix-algebra_FO1309 | 133 | 1.000 | x y | | |
| johnston-linear-matrix-algebra_FO1310 | 134 | 1.000 | E \in \mathcal{M}_{3} | | |
| johnston-linear-matrix-algebra_FO1311 | 134 | 0.975 | R_{1} \leftrightarrow R_{3} | | |
| johnston-linear-matrix-algebra_FO1312 | 134 | 1.000 | R_{1}+3 R_{2} | | |
| johnston-linear-matrix-algebra_FO1313 | 134 | 1.000 | R_{3}-4 R_{1} | | |
| johnston-linear-matrix-algebra_FO1314 | 134 | 1.000 | 3 R_{1} | | |
| johnston-linear-matrix-algebra_FO1315 | 134 | 1.000 | 6 R_{3} | | |
| johnston-linear-matrix-algebra_FO1316 | 134 | 0.995 | R_{2}-2 R_{3} | | |
| johnston-linear-matrix-algebra_FO1317 | 134 | 1.000 | R_{2}+3 R_{1} | | |
| johnston-linear-matrix-algebra_FO1318 | 134 | 1.000 | \left[\begin{array}{ll}6 & 3 \\ 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1319 | 134 | 0.892 | \left[\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1320 | 134 | 0.995 | \left[\begin{array}{lll}1 & 2 & 1 \\ 0 & 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1321 | 134 | 0.887 | \left[\begin{array}{ccc}2 & 4 & 0 \\ 1 & -2 & 0 \\ 2 & 0 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1322 | 134 | 0.997 | \left[\begin{array}{ccc}2 & 6 & 1 \\ 0 & 0 & 0 \\ 3 & -2 & 7\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1323 | 134 | 0.980 | \left[\begin{array}{ll}1 & 0 \\ 0 & 2 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1324 | 134 | 1.000 | \left[\begin{array}{ccc}1 & -2 & -2 \\ -2 & 1 & -2 \\ -2 & -2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1325 | 134 | 0.997 | \left[\begin{array}{llll}0 & 1 & 1 & 2 \\ 4 & 2 & 3 & 0 \\ 3 & 1 & 5 & 5 \\ 5 & 0 & 5 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1326 | 134 | 1.000 | \left[\begin{array}{cccc}-1 & -2 & 2 & 2 \\ -1 & 1 & 1 & 0 \\ 3 & 3 & 5 & 1 \\ 5 & 2 & 2 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1327 | 134 | 1.000 | \left[\begin{array}{ccccc}-1 & -3 & 3 & 5 & 4 \\ 1 & 5 & -2 & 4 & 5 \\ 5 & -3 & 0 & 1 & 0 \\ 4 & 1 & -2 & 1 & 1 \\ 4 & 4 & 5 & 3 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1328 | 134 | 1.000 | \left[\begin{array}{cccccc}-1 & -2 & -1 & 1 & 2 & 4 \\ -1 & -1 & 0 & 0 & -1 & 0 \\ 2 & 4 & 1 & 1 & -4 & -6 \\ 1 & 2 & 1 & 0 & -2 & -5 \\ -3 & -6 & -1 & -4 & 7 & 9 \\ 0 & -1 & 1 & -5 & 3 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1329 | 134 | 1.000 | E A | | |
| johnston-linear-matrix-algebra_FO1330 | 134 | 1.000 | \left[\begin{array}{cc}2 & -1 \\ -4 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1331 | 134 | 0.897 | \left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1332 | 134 | 0.992 | \left[\begin{array}{ccc}0 & -1 & 3 \\ 0 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1333 | 134 | 0.876 | \left[\begin{array}{cc}2 & 1 \\ -1 & 2 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1334 | 134 | 1.000 | \left[\begin{array}{ccc}1 & 2 & 1 \\ -1 & 0 & -1 \\ 0 & 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1335 | 134 | 0.999 | \left[\begin{array}{ccc}0 & 1 & 3 \\ 3 & -2 & 3 \\ 1 & 0 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1336 | 134 | 1.000 | \left[\begin{array}{cccc}4 & 0 & 2 & 4 \\ -1 & -1 & 1 & 2 \\ 5 & 0 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1337 | 134 | 1.000 | A^{6}=I | | |
| johnston-linear-matrix-algebra_FO1338 | 134 | 1.000 | A^{7}=O | | |
| johnston-linear-matrix-algebra_FO1339 | 134 | 1.000 | X A=B | | |
| johnston-linear-matrix-algebra_FO1340 | 134 | 1.000 | X=A^{-1} B | | |
| johnston-linear-matrix-algebra_FO1341 | 134 | 1.000 | a \neq 0 | | |
| johnston-linear-matrix-algebra_FO1342 | 134 | 1.000 | a=b | | |
| johnston-linear-matrix-algebra_FO1343 | 134 | 1.000 | a \neq b | | |
| johnston-linear-matrix-algebra_FO1344 | 134 | 1.000 | A^{-4} | | |
| johnston-linear-matrix-algebra_FO1345 | 134 | 1.000 | A^{k}=O | | |
| johnston-linear-matrix-algebra_FO1346 | 135 | 1.000 | I-A | | |
| johnston-linear-matrix-algebra_FO1347 | 135 | 1.000 | I+A | | |
| johnston-linear-matrix-algebra_FO1348 | 135 | 1.000 | A^{3}=O | | |
| johnston-linear-matrix-algebra_FO1349 | 135 | 1.000 | I+A+A^{2} | | |
| johnston-linear-matrix-algebra_FO1350 | 135 | 1.000 | P=A\left(A^{T} A\right)^{-1} A^{T} | | |
| johnston-linear-matrix-algebra_FO1351 | 135 | 1.000 | P | | |
| johnston-linear-matrix-algebra_FO1352 | 135 | 1.000 | P=P^{T} | | |
| johnston-linear-matrix-algebra_FO1353 | 135 | 0.999 | P^{2}=P | | |
| johnston-linear-matrix-algebra_FO1354 | 135 | 1.000 | P^{T}=P^{2}=P | | |
| johnston-linear-matrix-algebra_FO1355 | 135 | 0.997 | P, Q \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO1356 | 135 | 0.996 | [P \mid Q] | | |
| johnston-linear-matrix-algebra_FO1357 | 135 | 1.000 | \left[I \mid P^{-1} Q\right] | | |
| johnston-linear-matrix-algebra_FO1358 | 135 | 1.000 | A X=B | | |
| johnston-linear-matrix-algebra_FO1359 | 135 | 1.000 | A X B=O | | |
| johnston-linear-matrix-algebra_FO1360 | 135 | 1.000 | A X B=I | | |
| johnston-linear-matrix-algebra_FO1361 | 135 | 1.000 | A X+B X=A-B | | |
| johnston-linear-matrix-algebra_FO1362 | 135 | 1.000 | B^{-1} | | |
| johnston-linear-matrix-algebra_FO1363 | 135 | 0.996 | A_{1}, A_{2}, \ldots, A_{n} | | |
| johnston-linear-matrix-algebra_FO1364 | 135 | 0.959 | \left[\begin{array}{cc}A & I \\ I & O\end{array}\right]^{-1}=\left[\begin{array}{cc}O & I \\ I & -A\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1365 | 135 | 0.998 | \left[\begin{array}{cc}I & B \\ O & I\end{array}\right]^{-1}=\left[\begin{array}{cc}I & -B \\ O & I\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1366 | 135 | 0.946 | \left[\begin{array}{ll}A & B \\ O & D\end{array}\right]^{-1}=\left[\begin{array}{cc}A^{-1} & -A^{-1} B D^{-1} \\ O & D^{-1}\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1367 | 135 | 1.000 | S=D-C A^{-1} B | | |
| johnston-linear-matrix-algebra_FO1368 | 135 | 1.000 | \left[\begin{array}{llll}1 & 2 & 1 & 0 \\ 2 & 3 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1369 | 135 | 1.000 | \left[\begin{array}{llll}1 & 0 & 4 & 6 \\ 0 & 1 & 1 & 5 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1370 | 135 | 0.999 | \left[\begin{array}{ccccc}1 & 1 & 2 & -1 & 0 \\ 1 & 2 & 2 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1371 | 135 | 1.000 | \left[\begin{array}{llll}2 & 1 & 1 & 0 \\ 1 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 0 & 1 & 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1372 | 136 | 1.000 | A+\mathbf{v w}^{T} | | |
| johnston-linear-matrix-algebra_FO1373 | 136 | 0.520 | A_{n} | | |
| johnston-linear-matrix-algebra_FO1374 | 136 | 1.000 | A_{n}^{-1} | | |
| johnston-linear-matrix-algebra_FO1375 | 136 | 1.000 | n= | | |
| johnston-linear-matrix-algebra_FO1376 | 136 | 0.998 | \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1377 | 136 | 1.000 | \mathbf{v}, \mathbf{w} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1378 | 136 | 1.000 | \mathbf{v}+\mathbf{w} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1379 | 136 | 1.000 | \mathbf{v} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1380 | 136 | 1.000 | c \mathbf{v} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1382 | 137 | 0.993 | 1 / 2 | | |
| johnston-linear-matrix-algebra_FO1383 | 137 | 1.000 | x+y-3 z=0 | | |
| johnston-linear-matrix-algebra_FO1384 | 137 | 0.995 | (2,0) | | |
| johnston-linear-matrix-algebra_FO1385 | 137 | 0.995 | 2(2,0)=(4,0) | | |
| johnston-linear-matrix-algebra_FO1386 | 138 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO1387 | 138 | 1.000 | \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO1388 | 138 | 0.600 | \ldots, v_{n} | | |
| johnston-linear-matrix-algebra_FO1389 | 138 | 0.999 | \{(x, y) \in | | |
| johnston-linear-matrix-algebra_FO1390 | 138 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1391 | 138 | 1.000 | m \leq n | | |
| johnston-linear-matrix-algebra_FO1392 | 138 | 0.993 | \mathbb{R}^{1} | | |
| johnston-linear-matrix-algebra_FO1393 | 138 | 1.000 | y=x^{2} | | |
| johnston-linear-matrix-algebra_FO1394 | 138 | 1.000 | \left\{(x, y) \in \mathbb{R}^{2}: x \geq 0, y \geq 0\right\} | | |
| johnston-linear-matrix-algebra_FO1395 | 138 | 1.000 | \mathbf{v}= | | |
| johnston-linear-matrix-algebra_FO1396 | 138 | 0.867 | -1,1 | | |
| johnston-linear-matrix-algebra_FO1397 | 138 | 0.867 | \mathbf{v}+\mathbf{w}=(0,2) | | |
| johnston-linear-matrix-algebra_FO1398 | 138 | 1.000 | 2=0^{2} | | |
| johnston-linear-matrix-algebra_FO1399 | 139 | 1.000 | \{\mathbf{0}\} | | |
| johnston-linear-matrix-algebra_FO1400 | 139 | 1.000 | \mathbf{b} \neq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1401 | 139 | 1.000 | \operatorname{ker}(A) | | |
| johnston-linear-matrix-algebra_FO1402 | 139 | 1.000 | \mathbf{v}=(1,1) | | |
| johnston-linear-matrix-algebra_FO1403 | 139 | 1.000 | -\mathbf{v}=(-1,-1) | | |
| johnston-linear-matrix-algebra_FO1404 | 139 | 1.000 | v_{1}+v_{2}-3 v_{3}=0 | | |
| johnston-linear-matrix-algebra_FO1405 | 139 | 1.000 | w_{1}+w_{2}-3 w_{3}=0 | | |
| johnston-linear-matrix-algebra_FO1406 | 139 | 1.000 | A \mathbf{x} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO1407 | 139 | 1.000 | A \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO1408 | 139 | 0.751 | \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1409 | 139 | 1.000 | \operatorname{null}(A) | | |
| johnston-linear-matrix-algebra_FO1410 | 139 | 1.000 | A \mathbf{0}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1411 | 139 | 1.000 | \mathbf{0} \in \operatorname{null}(A) | | |
| johnston-linear-matrix-algebra_FO1412 | 140 | 1.000 | \mathbf{v}, \mathbf{w} \in \operatorname{null}(A) | | |
| johnston-linear-matrix-algebra_FO1413 | 140 | 1.000 | A \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1414 | 140 | 1.000 | A \mathbf{w}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1415 | 140 | 1.000 | \mathbf{v}+\mathbf{w} \in \operatorname{null}(A) | | |
| johnston-linear-matrix-algebra_FO1416 | 140 | 1.000 | \mathbf{v} \in \operatorname{null}(A) | | |
| johnston-linear-matrix-algebra_FO1417 | 140 | 0.984 | c \mathbf{v} \in \operatorname{null}(A) | | |
| johnston-linear-matrix-algebra_FO1418 | 140 | 1.000 | A \mathbf{0}=\mathbf{0} \in \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1419 | 140 | 1.000 | A \mathbf{x}, A \mathbf{y} \in \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1420 | 140 | 1.000 | A \mathbf{x}+A \mathbf{y}=A(\mathbf{x}+\mathbf{y}) \in \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1421 | 140 | 1.000 | A(c \mathbf{x})=c(A \mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO1422 | 140 | 0.899 | A=\left[\begin{array}{ll}2 & -2 \\ 1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1423 | 140 | 1.000 | \mathbf{x}=(x, y) | | |
| johnston-linear-matrix-algebra_FO1424 | 140 | 1.000 | x-y | | |
| johnston-linear-matrix-algebra_FO1425 | 140 | 1.000 | z=x-y | | |
| johnston-linear-matrix-algebra_FO1426 | 140 | 1.000 | A \mathbf{x}=(2 z, z)=z(2,1) | | |
| johnston-linear-matrix-algebra_FO1427 | 140 | 1.000 | x=y | | |
| johnston-linear-matrix-algebra_FO1428 | 140 | 1.000 | \mathbf{x}=(x, x)=x(1,1) | | |
| johnston-linear-matrix-algebra_FO1429 | 141 | 1.000 | \mathbf{v}_{1}, \ldots, \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO1430 | 141 | 1.000 | c_{1} \mathbf{v}_{1}+\cdots+c_{k} \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO1431 | 141 | 0.724 | 2(2,1)=(4,2) | | |
| johnston-linear-matrix-algebra_FO1432 | 141 | 1.000 | (4,2) | | |
| johnston-linear-matrix-algebra_FO1433 | 141 | 1.000 | \{(2,1),(4,2)\} | | |
| johnston-linear-matrix-algebra_FO1435 | 141 | 1.000 | B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO1436 | 141 | 0.511 | ((2,1)) | | |
| johnston-linear-matrix-algebra_FO1437 | 141 | 1.000 | \operatorname{span}\left(\mathbf{e}_{1}, \mathbf{e}_{2}\right)=\mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO1438 | 141 | 1.000 | \operatorname{span}\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right)=\mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1439 | 142 | 1.000 | \operatorname{span}((1,2),(2,1)) | | |
| johnston-linear-matrix-algebra_FO1440 | 142 | 1.000 | \operatorname{span}((1,2,1),(2,1,1)) | | |
| johnston-linear-matrix-algebra_FO1441 | 142 | 1.000 | \operatorname{span}((1,2),(2,1))=\mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO1442 | 142 | 1.000 | (x, y) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO1443 | 142 | 0.726 | \left[\begin{array}{cc|c}0 & 0 & b\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1444 | 142 | 1.000 | (x, y) | | |
| johnston-linear-matrix-algebra_FO1445 | 143 | 1.000 | \operatorname{span}\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO1446 | 143 | 1.000 | (x, y, z) \in \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO1447 | 143 | 1.000 | z=\frac{1}{3} x+\frac{1}{3} y | | |
| johnston-linear-matrix-algebra_FO1448 | 143 | 0.955 | (x, y, z) | | |
| johnston-linear-matrix-algebra_FO1449 | 143 | 0.703 | (2,1,1) | | |
| johnston-linear-matrix-algebra_FO1450 | 143 | 1.000 | \mathbf{v}, \mathbf{w} \in \operatorname{span}\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO1451 | 143 | 0.991 | c_{1}, c_{2}, \ldots, c_{k}, d_{1}, d_{2}, \ldots, d_{k} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1452 | 143 | 1.000 | \mathbf{v}+\mathbf{w} \in \operatorname{span}\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO1453 | 144 | 1.000 | \mathcal{S}=\operatorname{span}\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO1454 | 144 | 1.000 | \left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO1455 | 144 | 1.000 | \operatorname{col}(A) | | |
| johnston-linear-matrix-algebra_FO1456 | 144 | 0.997 | c \mathbf{v} \in \operatorname{span}\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO1457 | 144 | 0.982 | (2,3) | | |
| johnston-linear-matrix-algebra_FO1458 | 144 | 1.000 | \operatorname{span}((1,2,3),(3,-1,2)) | | |
| johnston-linear-matrix-algebra_FO1459 | 144 | 1.000 | \operatorname{span}\left(\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}\right) | | |
| johnston-linear-matrix-algebra_FO1460 | 145 | 0.958 | A \operatorname{span} \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1461 | 145 | 0.755 | \operatorname{span} \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1462 | 146 | 1.000 | \left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right\} \subseteq \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1463 | 146 | 1.000 | -x+y=-2 | | |
| johnston-linear-matrix-algebra_FO1464 | 146 | 1.000 | x-y=2 | | |
| johnston-linear-matrix-algebra_FO1465 | 146 | 1.000 | \{(2,3),(1,0),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO1466 | 146 | 1.000 | \{(1,0,0),(0,1,0),(0,0,1)\} | | |
| johnston-linear-matrix-algebra_FO1467 | 146 | 1.000 | c_{1}, c_{2}, \ldots, c_{k} | | |
| johnston-linear-matrix-algebra_FO1468 | 146 | 0.998 | \{(1,-1,0),(-2,1,2),(1,1,-4)\} | | |
| johnston-linear-matrix-algebra_FO1469 | 146 | 1.000 | \{(1,2,3),(1,0,1),(0,-1,2)\} | | |
| johnston-linear-matrix-algebra_FO1470 | 147 | 1.000 | c_{3} | | |
| johnston-linear-matrix-algebra_FO1471 | 147 | 1.000 | c_{1}, c_{2}, c_{3} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1472 | 147 | 1.000 | c_{3}=1 | | |
| johnston-linear-matrix-algebra_FO1473 | 147 | 1.000 | c_{2}=2 | | |
| johnston-linear-matrix-algebra_FO1474 | 147 | 1.000 | c_{1}=3 | | |
| johnston-linear-matrix-algebra_FO1475 | 147 | 1.000 | c_{1}=c_{2}=c_{3}=0 | | |
| johnston-linear-matrix-algebra_FO1476 | 148 | 0.997 | \left\{\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}\right\} | | |
| johnston-linear-matrix-algebra_FO1477 | 148 | 0.998 | A=\left[\mathbf{a}_{1}\left|\mathbf{a}_{2}\right| \cdots \mid \mathbf{a}_{n}\right] | | |
| johnston-linear-matrix-algebra_FO1478 | 148 | 0.778 | x_{1}, x_{2}, \ldots, x_{n}( | | |
| johnston-linear-matrix-algebra_FO1479 | 148 | 0.778 | x_{1} \mathbf{a}_{1}+x_{2} \mathbf{a}_{2}+\cdots+x_{n} \mathbf{a}_{n}= | | |
| johnston-linear-matrix-algebra_FO1480 | 148 | 0.999 | \mathbf{v}_{1} | | |
| johnston-linear-matrix-algebra_FO1481 | 148 | 1.000 | \{(1,2,1),(2,4,2),(2,1,1)\} | | |
| johnston-linear-matrix-algebra_FO1482 | 149 | 0.863 | \{(1,2),(3,1)\} | | |
| johnston-linear-matrix-algebra_FO1483 | 149 | 0.985 | \{(1,2,3),(2,4,6)\} | | |
| johnston-linear-matrix-algebra_FO1484 | 149 | 0.861 | (3,1)=c(1,2) | | |
| johnston-linear-matrix-algebra_FO1485 | 150 | 1.000 | (2,4,6)=2(1,2,3) | | |
| johnston-linear-matrix-algebra_FO1486 | 150 | 1.000 | (n-1) | | |
| johnston-linear-matrix-algebra_FO1487 | 151 | 1.000 | a_{0}=1 | | |
| johnston-linear-matrix-algebra_FO1488 | 151 | 1.000 | a_{1}=2 | | |
| johnston-linear-matrix-algebra_FO1489 | 151 | 1.000 | a_{2}=3 | | |
| johnston-linear-matrix-algebra_FO1490 | 151 | 1.000 | V | | |
| johnston-linear-matrix-algebra_FO1491 | 151 | 1.000 | (n+1) \times(n+1) | | |
| johnston-linear-matrix-algebra_FO1492 | 151 | 1.000 | x_{0}, x_{1}, \ldots, x_{n} | | |
| johnston-linear-matrix-algebra_FO1493 | 151 | 1.000 | a_{0}, a_{1}, \ldots, a_{n} | | |
| johnston-linear-matrix-algebra_FO1494 | 151 | 0.998 | a_{i} \neq a_{j} | | |
| johnston-linear-matrix-algebra_FO1495 | 151 | 0.989 | c_{0}=c_{1}=\cdots=c_{n}=0 | | |
| johnston-linear-matrix-algebra_FO1496 | 151 | 1.000 | p(x)=c_{0}+c_{1} x+c_{2} x^{2}+\cdots+c_{n} x^{n} | | |
| johnston-linear-matrix-algebra_FO1497 | 151 | 1.000 | p\left(a_{0}\right)=0 | | |
| johnston-linear-matrix-algebra_FO1498 | 151 | 1.000 | p\left(a_{1}\right)=0 | | |
| johnston-linear-matrix-algebra_FO1499 | 151 | 1.000 | p\left(a_{n}\right)=0 | | |
| johnston-linear-matrix-algebra_FO1500 | 151 | 1.000 | n+1 | | |
| johnston-linear-matrix-algebra_FO1501 | 151 | 0.982 | c_{0}=c_{1}= | | |
| johnston-linear-matrix-algebra_FO1502 | 151 | 1.000 | \cdots=c_{n}=0 | | |
| johnston-linear-matrix-algebra_FO1503 | 151 | 1.000 | \left(x_{0}, y_{0}\right),\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right) | | |
| johnston-linear-matrix-algebra_FO1504 | 151 | 0.960 | p\left(x_{0}\right)=y_{0}, p\left(x_{1}\right)=y_{1}, \ldots, p\left(x_{n}\right)=y_{n} | | |
| johnston-linear-matrix-algebra_FO1505 | 151 | 1.000 | n=1 | | |
| johnston-linear-matrix-algebra_FO1506 | 152 | 1.000 | p\left(x_{0}\right)=y_{0} | | |
| johnston-linear-matrix-algebra_FO1507 | 152 | 0.623 | c_{n} x_{0}^{n}+\cdots+c_{0}=y_{0} | | |
| johnston-linear-matrix-algebra_FO1508 | 152 | 1.000 | c_{0}, c_{1}, \ldots, c_{n} | | |
| johnston-linear-matrix-algebra_FO1509 | 152 | 1.000 | p\left(x_{1}\right)=y_{1}, \ldots, p\left(x_{n}\right)=y_{n} | | |
| johnston-linear-matrix-algebra_FO1510 | 153 | 1.000 | \left(c_{0}, c_{1}, c_{2}\right)=(-1,-3,2) | | |
| johnston-linear-matrix-algebra_FO1511 | 153 | 1.000 | p(-2)=13 | | |
| johnston-linear-matrix-algebra_FO1512 | 153 | 1.000 | p(1)=-2 | | |
| johnston-linear-matrix-algebra_FO1513 | 153 | 1.000 | p(3)=8 | | |
| johnston-linear-matrix-algebra_FO1514 | 153 | 1.000 | y=x+1 | | |
| johnston-linear-matrix-algebra_FO1515 | 153 | 1.000 | y=\sin (x) | | |
| johnston-linear-matrix-algebra_FO1516 | 153 | 0.995 | x+2 y+3 z=4 | | |
| johnston-linear-matrix-algebra_FO1517 | 153 | 1.000 | x-y+8 z=0 | | |
| johnston-linear-matrix-algebra_FO1518 | 153 | 1.000 | \left\{(x, y) \in \mathbb{R}^{2}: x+2 y=0\right\} | | |
| johnston-linear-matrix-algebra_FO1519 | 153 | 1.000 | \left\{(x, y) \in \mathbb{R}^{2}: x+y \geq 0\right\} | | |
| johnston-linear-matrix-algebra_FO1520 | 153 | 1.000 | \left\{(x, y) \in \mathbb{R}^{2}: x y \geq 0\right\} | | |
| johnston-linear-matrix-algebra_FO1521 | 153 | 1.000 | \left\{(x, y, z) \in \mathbb{R}^{3}: x y+y z=0\right\} | | |
| johnston-linear-matrix-algebra_FO1522 | 153 | 0.995 | \{(2,3),(0,0)\} | | |
| johnston-linear-matrix-algebra_FO1523 | 153 | 1.000 | \{(0,1,-1),(1,2,1),(3,-1,4)\} | | |
| johnston-linear-matrix-algebra_FO1524 | 153 | 1.000 | \{(1,2,1),(0,1,-1),(2,5,1)\} | | |
| johnston-linear-matrix-algebra_FO1525 | 153 | 1.000 | \{(1,1,0),(1,0,-1),(0,1,1),(1,2,1)\} | | |
| johnston-linear-matrix-algebra_FO1526 | 153 | 0.359 | *(\mathbf{a})\left[\begin{array}{rr}1 & 1 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1527 | 153 | 0.359 | *(\mathbf{c}) \quad\left[\begin{array}{ll}0 & 1 \\ 2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1528 | 153 | 0.083 | \left.\begin{array}{r}{[11} \\ 22 \\ \text { (d) }\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1529 | 153 | 0.420 | *(\mathbf{a}) \quad\left[\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1530 | 153 | 0.420 | *(\mathbf{c}) \quad\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1531 | 153 | 1.000 | \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1532 | 153 | 0.999 | \left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1533 | 153 | 1.000 | \{(1,0,1),(1,1,1)\} | | |
| johnston-linear-matrix-algebra_FO1534 | 153 | 0.640 | \{(1,0,-1),(1,1,1),(1,2,-1)\} | | |
| johnston-linear-matrix-algebra_FO1535 | 153 | 1.000 | \{(1,2,3),(4,5,6),(7,8,9)\} | | |
| johnston-linear-matrix-algebra_FO1536 | 153 | 0.969 | \{(1,1),(2,1),(3,-2)\} | | |
| johnston-linear-matrix-algebra_FO1537 | 153 | 0.990 | \{(2,1,0),(0,0,0),(1,1,2)\} | | |
| johnston-linear-matrix-algebra_FO1538 | 153 | 0.516 | \{(1,2,4,1),(2,4,-1,3),(-1,1,1,-1)\} | | |
| johnston-linear-matrix-algebra_FO1539 | 153 | 1.000 | \{(0,1,1,1),(1,0,1,1),(1,1,0,1),(1,1,1,0)\} | | |
| johnston-linear-matrix-algebra_FO1540 | 153 | 0.963 | \{(4,2,5,2),(3,1,2,4),(1,4,2,3),(3,1,4,2)\} | | |
| johnston-linear-matrix-algebra_FO1541 | 153 | 0.998 | \{(4,4,4,3),(3,3,-1,1),(-1,2,1,2) | | |
| johnston-linear-matrix-algebra_FO1542 | 153 | 0.999 | \{(2,-1,4,-1),(3,1,2,1),(0,1,3,4) | | |
| johnston-linear-matrix-algebra_FO1543 | 153 | 1.000 | \{(3,5,1,4),(4,4,5,5),(5,0,4,3),(1,1,5,-1)\} | | |
| johnston-linear-matrix-algebra_FO1544 | 153 | 1.000 | \{(5,4,5,1,5),(4,3,3,0,4),(-1,0,3,-1,4)\} | | |
| johnston-linear-matrix-algebra_FO1545 | 153 | 0.999 | \{(5,-1,2,4,3),(4,-8,1,-4,-9),(2,2,1,4,5)\} | | |
| johnston-linear-matrix-algebra_FO1546 | 154 | 1.000 | \mathbf{v}_{2} | | |
| johnston-linear-matrix-algebra_FO1547 | 154 | 1.000 | \mathbf{v}_{3} | | |
| johnston-linear-matrix-algebra_FO1548 | 154 | 0.999 | \operatorname{span}(\mathbf{v}, \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO1549 | 154 | 1.000 | k \geq n | | |
| johnston-linear-matrix-algebra_FO1550 | 154 | 1.000 | \operatorname{span}\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right)=\mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1551 | 154 | 0.534 | (k,-1,4) | | |
| johnston-linear-matrix-algebra_FO1552 | 154 | 0.976 | (1,2,3),(3, k, k+3),(2,4, k) | | |
| johnston-linear-matrix-algebra_FO1553 | 154 | 0.870 | (5,7) | | |
| johnston-linear-matrix-algebra_FO1554 | 154 | 0.721 | (4,4) | | |
| johnston-linear-matrix-algebra_FO1555 | 154 | 0.772 | (4,10) | | |
| johnston-linear-matrix-algebra_FO1556 | 154 | 0.999 | 1,2, \ldots, n^{2} | | |
| johnston-linear-matrix-algebra_FO1557 | 154 | 1.000 | n \leq 2 | | |
| johnston-linear-matrix-algebra_FO1558 | 154 | 1.000 | \mathbf{v}_{1}, \ldots, \mathbf{v}_{k} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1559 | 154 | 1.000 | A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1560 | 154 | 1.000 | \{\mathbf{v}, A \mathbf{v}\} | | |
| johnston-linear-matrix-algebra_FO1561 | 154 | 1.000 | \mathbf{v}=(1,2,3,4,5) | | |
| johnston-linear-matrix-algebra_FO1562 | 154 | 1.000 | A \mathbf{v}=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO1563 | 154 | 1.000 | \{\mathbf{v}, \mathbf{w}\} \subset \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1564 | 154 | 1.000 | \{\mathbf{v}, \mathbf{w}, \mathbf{x}\} \subseteq \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1565 | 154 | 1.000 | \{\mathbf{v}+\mathbf{w}, \mathbf{v}+\mathbf{x}, \mathbf{w}-\mathbf{x}\} | | |
| johnston-linear-matrix-algebra_FO1566 | 154 | 1.000 | \{\mathbf{v}+\mathbf{w}, \mathbf{v}+\mathbf{x}, \mathbf{w}+\mathbf{x}\} | | |
| johnston-linear-matrix-algebra_FO1567 | 154 | 1.000 | \{\mathbf{v}, \mathbf{v}+\mathbf{w}, \mathbf{v}+\mathbf{w}+\mathbf{x}\} | | |
| johnston-linear-matrix-algebra_FO1568 | 154 | 1.000 | B \subseteq \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1569 | 154 | 0.822 | \mathbf{0} \in B | | |
| johnston-linear-matrix-algebra_FO1570 | 154 | 1.000 | \mathbf{w}, \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1571 | 154 | 1.000 | B \subseteq C \subseteq \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1572 | 154 | 1.000 | n>m | | |
| johnston-linear-matrix-algebra_FO1573 | 155 | 0.943 | (A B) \subseteq \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1574 | 155 | 1.000 | \operatorname{null}(B) \subseteq \operatorname{null}(A B) | | |
| johnston-linear-matrix-algebra_FO1575 | 155 | 1.000 | (A B) | | |
| johnston-linear-matrix-algebra_FO1576 | 155 | 1.000 | \operatorname{range}(B) | | |
| johnston-linear-matrix-algebra_FO1577 | 155 | 0.988 | \operatorname{null}(A B) | | |
| johnston-linear-matrix-algebra_FO1578 | 155 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1579 | 155 | 1.000 | c_{1}, c_{2}, \ldots, c_{n} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1580 | 155 | 1.000 | \left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\} | | |
| johnston-linear-matrix-algebra_FO1581 | 155 | 1.000 | \left\{c_{1} \mathbf{v}_{1}, c_{2} \mathbf{v}_{2}, \ldots, c_{n} \mathbf{v}_{n}\right\} | | |
| johnston-linear-matrix-algebra_FO1582 | 155 | 0.402 | V_{n} | | |
| johnston-linear-matrix-algebra_FO1583 | 155 | 1.000 | V_{n}^{-1} | | |
| johnston-linear-matrix-algebra_FO1584 | 155 | 0.812 | (3,3,2)= | | |
| johnston-linear-matrix-algebra_FO1585 | 155 | 0.898 | (1,2,1)+(2,1,1) | | |
| johnston-linear-matrix-algebra_FO1586 | 155 | 0.993 | \operatorname{span}((1,2,1),(2,1,1),(3,3,2)) | | |
| johnston-linear-matrix-algebra_FO1588 | 156 | 0.988 | \{\mathbf{v}, \mathbf{w}\} | | |
| johnston-linear-matrix-algebra_FO1589 | 156 | 1.000 | \mathcal{S} \subseteq \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1590 | 156 | 1.000 | \left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right\} \subset \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1591 | 156 | 1.000 | \left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right\} | | |
| johnston-linear-matrix-algebra_FO1592 | 156 | 0.742 | \left(c_{1}, c_{2}, \ldots, c_{n}\right) | | |
| johnston-linear-matrix-algebra_FO1593 | 156 | 0.999 | c_{1}=c_{2}=\cdots=c_{n}=0 | | |
| johnston-linear-matrix-algebra_FO1594 | 156 | 0.999 | B=\{(1,2,1),(2,1,1)\}, \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1595 | 156 | 1.000 | B=\{(1,1,2),(1,2,1),(2,1,1)\}, \mathcal{S}=\mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO1596 | 156 | 1.000 | \operatorname{span}(B)=\mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO1597 | 157 | 0.973 | \} | | |
| johnston-linear-matrix-algebra_FO1598 | 157 | 0.999 | B=\{ \} | | |
| johnston-linear-matrix-algebra_FO1599 | 157 | 1.000 | \left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right\} | | |
| johnston-linear-matrix-algebra_FO1600 | 157 | 1.000 | \{(1,1,2),(1,2,1),(2,1,1)\} | | |
| johnston-linear-matrix-algebra_FO1601 | 158 | 1.000 | |B| | | |
| johnston-linear-matrix-algebra_FO1603 | 158 | 1.000 | B, C \subseteq \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1604 | 158 | 1.000 | \operatorname{span}(C)=\mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1605 | 158 | 1.000 | B= | | |
| johnston-linear-matrix-algebra_FO1606 | 158 | 1.000 | C=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{\ell}\right\} | | |
| johnston-linear-matrix-algebra_FO1607 | 158 | 1.000 | k=|B| | | |
| johnston-linear-matrix-algebra_FO1608 | 158 | 1.000 | \ell=|C| | | |
| johnston-linear-matrix-algebra_FO1609 | 158 | 1.000 | \ell<k | | |
| johnston-linear-matrix-algebra_FO1610 | 158 | 1.000 | \mathbf{v}_{i} \in B \subseteq \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1611 | 158 | 1.000 | 1 \leq i \leq k | | |
| johnston-linear-matrix-algebra_FO1612 | 158 | 1.000 | \mathbf{v}_{i} | | |
| johnston-linear-matrix-algebra_FO1613 | 158 | 1.000 | \mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{\ell} | | |
| johnston-linear-matrix-algebra_FO1614 | 158 | 0.999 | (1 \leq i \leq k, 1 \leq j \leq \ell) | | |
| johnston-linear-matrix-algebra_FO1615 | 158 | 1.000 | A \in \mathcal{M}_{k, \ell} | | |
| johnston-linear-matrix-algebra_FO1616 | 158 | 0.997 | a_{i, j}, V \in \mathcal{M}_{n, k} | | |
| johnston-linear-matrix-algebra_FO1617 | 158 | 1.000 | W \in \mathcal{M}_{n, \ell} | | |
| johnston-linear-matrix-algebra_FO1618 | 158 | 1.000 | V=W A^{T} | | |
| johnston-linear-matrix-algebra_FO1619 | 158 | 1.000 | W A^{T} | | |
| johnston-linear-matrix-algebra_FO1620 | 158 | 1.000 | \mathbf{x} \in \mathbb{R}^{k} | | |
| johnston-linear-matrix-algebra_FO1621 | 158 | 1.000 | A^{T} \mathbf{x}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1622 | 159 | 1.000 | V \mathbf{x}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1623 | 159 | 1.000 | B \subseteq \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1624 | 159 | 1.000 | C \subseteq \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1625 | 159 | 1.000 | |B| \leq|C| | | |
| johnston-linear-matrix-algebra_FO1626 | 159 | 1.000 | |C| \leq|B| | | |
| johnston-linear-matrix-algebra_FO1627 | 159 | 1.000 | |B|=|C| | | |
| johnston-linear-matrix-algebra_FO1628 | 159 | 1.000 | \operatorname{dim}(\mathcal{S}) | | |
| johnston-linear-matrix-algebra_FO1629 | 159 | 1.000 | \operatorname{dim}\left(\mathbb{R}^{n}\right)=n | | |
| johnston-linear-matrix-algebra_FO1630 | 160 | 0.994 | \leq \operatorname{dim}(\mathcal{S}) \leq | | |
| johnston-linear-matrix-algebra_FO1631 | 160 | 1.000 | \mathcal{S}=\operatorname{span}((1,1,1),(1,2,3),(3,2,1)) | | |
| johnston-linear-matrix-algebra_FO1632 | 160 | 0.998 | 2 x-y+3 z=0 | | |
| johnston-linear-matrix-algebra_FO1633 | 160 | 1.000 | B=\{(1,1,1),(1,2,3),(3,2,1)\} | | |
| johnston-linear-matrix-algebra_FO1634 | 160 | 1.000 | \mathcal{S}=\operatorname{span}(B) | | |
| johnston-linear-matrix-algebra_FO1635 | 160 | 0.671 | \operatorname{span}(B) | | |
| johnston-linear-matrix-algebra_FO1636 | 160 | 1.000 | C=\{(1,2,3),(3,2,1)\} | | |
| johnston-linear-matrix-algebra_FO1637 | 161 | 0.856 | (3,3,-1) \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1638 | 161 | 0.992 | 2(3)-(3)+3(-1)=0 | | |
| johnston-linear-matrix-algebra_FO1639 | 161 | 0.889 | (1,-1,-1) | | |
| johnston-linear-matrix-algebra_FO1640 | 161 | 0.889 | 2(1)-(-1)+ | | |
| johnston-linear-matrix-algebra_FO1641 | 161 | 1.000 | 3(-1)=0 | | |
| johnston-linear-matrix-algebra_FO1642 | 161 | 0.809 | \operatorname{span}((1,-1,-1)) | | |
| johnston-linear-matrix-algebra_FO1643 | 161 | 0.979 | (3,3,-1) | | |
| johnston-linear-matrix-algebra_FO1644 | 161 | 0.979 | B=\{(1,-1,-1),(3,3,-1)\} | | |
| johnston-linear-matrix-algebra_FO1645 | 161 | 0.655 | \operatorname{span}(B)=\mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1646 | 161 | 1.000 | \frac{2}{3} x-\frac{1}{3} y+z=0 | | |
| johnston-linear-matrix-algebra_FO1647 | 161 | 1.000 | \operatorname{dim}(\mathcal{S})=2 | | |
| johnston-linear-matrix-algebra_FO1648 | 161 | 1.000 | B \subseteq C | | |
| johnston-linear-matrix-algebra_FO1649 | 161 | 1.000 | C \subseteq B | | |
| johnston-linear-matrix-algebra_FO1650 | 161 | 1.000 | \operatorname{span}(B)= | | |
| johnston-linear-matrix-algebra_FO1651 | 162 | 1.000 | \mathbf{w} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1652 | 162 | 1.000 | \left\{\mathbf{w}, \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO1653 | 162 | 1.000 | d=c_{1}=c_{2}=\cdots=c_{k}=0 | | |
| johnston-linear-matrix-algebra_FO1654 | 162 | 1.000 | d=0 | | |
| johnston-linear-matrix-algebra_FO1655 | 162 | 0.999 | c_{1}=c_{2}=\cdots=c_{k}=0 | | |
| johnston-linear-matrix-algebra_FO1656 | 162 | 1.000 | k \neq \operatorname{dim}(\mathcal{S}) | | |
| johnston-linear-matrix-algebra_FO1657 | 162 | 1.000 | k=\operatorname{dim}(\mathcal{S}) | | |
| johnston-linear-matrix-algebra_FO1658 | 162 | 1.000 | \operatorname{dim}(\mathcal{S})=k | | |
| johnston-linear-matrix-algebra_FO1659 | 163 | 1.000 | B=\{(1,2,3),(4,1,2)\} | | |
| johnston-linear-matrix-algebra_FO1660 | 163 | 1.000 | B=\{(1,-1,-1),(2,2,-1),(1,1,0)\} | | |
| johnston-linear-matrix-algebra_FO1661 | 163 | 0.913 | B=\{(1,-1,-1),(2,2,-1)\} | | |
| johnston-linear-matrix-algebra_FO1662 | 163 | 1.000 | \mathcal{S} \subset \mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO1663 | 163 | 1.000 | 3 x-y+4 z=0 | | |
| johnston-linear-matrix-algebra_FO1664 | 163 | 0.624 | (2,2,-1) | | |
| johnston-linear-matrix-algebra_FO1665 | 164 | 0.993 | \mathcal{S}=\{\mathbf{0}\} | | |
| johnston-linear-matrix-algebra_FO1666 | 164 | 1.000 | (2,1,1)=c_{1}(1,2,1)+ | | |
| johnston-linear-matrix-algebra_FO1667 | 164 | 0.975 | c_{2}(1,-1,0) | | |
| johnston-linear-matrix-algebra_FO1668 | 164 | 1.000 | B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{m}\right\} | | |
| johnston-linear-matrix-algebra_FO1669 | 164 | 1.000 | \mathbf{w} \in \operatorname{span}(B) | | |
| johnston-linear-matrix-algebra_FO1670 | 164 | 1.000 | \left\{\mathbf{w}, \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{m}\right\} | | |
| johnston-linear-matrix-algebra_FO1671 | 164 | 1.000 | m+1 | | |
| johnston-linear-matrix-algebra_FO1672 | 164 | 1.000 | d, c_{1}, c_{2}, \ldots, c_{m} | | |
| johnston-linear-matrix-algebra_FO1673 | 164 | 1.000 | c_{1}=c_{2}=\cdots= | | |
| johnston-linear-matrix-algebra_FO1674 | 164 | 1.000 | c_{m}=0 | | |
| johnston-linear-matrix-algebra_FO1675 | 164 | 0.991 | d \neq 0 | | |
| johnston-linear-matrix-algebra_FO1676 | 164 | 1.000 | A=\left[\begin{array}{ccc}1 & 1 & 2 \\ 2 & -1 & 1 \\ 1 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1677 | 164 | 1.000 | (2,1,1)=(1,2,1)+(1,-1,0) | | |
| johnston-linear-matrix-algebra_FO1678 | 164 | 1.000 | \{(1,2,1),(1,-1,0)\} | | |
| johnston-linear-matrix-algebra_FO1679 | 165 | 1.000 | 2+1=3 | | |
| johnston-linear-matrix-algebra_FO1680 | 165 | 1.000 | x_{3} | | |
| johnston-linear-matrix-algebra_FO1681 | 165 | 1.000 | x_{1} | | |
| johnston-linear-matrix-algebra_FO1682 | 165 | 1.000 | x_{2} | | |
| johnston-linear-matrix-algebra_FO1683 | 165 | 1.000 | x_{1}+x_{3}=0 | | |
| johnston-linear-matrix-algebra_FO1684 | 165 | 1.000 | x_{1}=-x_{3} | | |
| johnston-linear-matrix-algebra_FO1685 | 165 | 1.000 | x_{2}+x_{3}=0 | | |
| johnston-linear-matrix-algebra_FO1686 | 165 | 1.000 | x_{2}=-x_{3} | | |
| johnston-linear-matrix-algebra_FO1687 | 165 | 1.000 | \left(x_{1}, x_{2}, x_{3}\right)= | | |
| johnston-linear-matrix-algebra_FO1688 | 165 | 1.000 | \left(-x_{3},-x_{3}, x_{3}\right)=x_{3}(-1,-1,1) | | |
| johnston-linear-matrix-algebra_FO1689 | 165 | 1.000 | \{(-1,-1,1)\} | | |
| johnston-linear-matrix-algebra_FO1690 | 165 | 1.000 | R \mathbf{x}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1691 | 165 | 0.837 | \operatorname{range}(R) | | |
| johnston-linear-matrix-algebra_FO1692 | 165 | 0.572 | (R) | | |
| johnston-linear-matrix-algebra_FO1693 | 166 | 0.984 | \{(1,1,-1,2),(0,1,0,1),(0,0,1,-1)\}) | | |
| johnston-linear-matrix-algebra_FO1694 | 166 | 1.000 | x_{5} | | |
| johnston-linear-matrix-algebra_FO1695 | 166 | 1.000 | x_{1}, x_{2} | | |
| johnston-linear-matrix-algebra_FO1696 | 166 | 1.000 | x_{4} | | |
| johnston-linear-matrix-algebra_FO1697 | 167 | 1.000 | \{(-1,1,1,0,0),(1,-2,0,-3,1)\} | | |
| johnston-linear-matrix-algebra_FO1698 | 167 | 0.977 | \operatorname{range}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1699 | 167 | 0.977 | \operatorname{null}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1700 | 168 | 0.719 | \left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1701 | 168 | 1.000 | \{(0,-1,1,1)\} | | |
| johnston-linear-matrix-algebra_FO1702 | 169 | 1.000 | \operatorname{row}(A) | | |
| johnston-linear-matrix-algebra_FO1703 | 169 | 1.000 | \mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{m} | | |
| johnston-linear-matrix-algebra_FO1704 | 169 | 1.000 | \operatorname{range}\left(A^{T}\right)=\operatorname{range}\left(R^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1705 | 169 | 0.999 | A^{T} E^{T}=R^{T} | | |
| johnston-linear-matrix-algebra_FO1706 | 169 | 1.000 | E^{T} | | |
| johnston-linear-matrix-algebra_FO1707 | 169 | 1.000 | \mathbf{v}_{1}, \ldots, \mathbf{v}_{m} | | |
| johnston-linear-matrix-algebra_FO1708 | 169 | 1.000 | R^{T} | | |
| johnston-linear-matrix-algebra_FO1709 | 169 | 1.000 | A^{T} \mathbf{v}_{m-k+1}=A^{T} \mathbf{v}_{m-k+2}=\cdots=A^{T} \mathbf{v}_{m}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1710 | 170 | 1.000 | A=\left[\begin{array}{cccc}1 & 1 & 1 & -1 \\ 0 & 1 & 1 & 0 \\ -1 & 1 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1711 | 170 | 1.000 | \operatorname{range}(A):\{(1,0,-1),(1,1,1)\} | | |
| johnston-linear-matrix-algebra_FO1712 | 170 | 1.000 | \operatorname{null}\left(A^{T}\right):\{(1,-2,1)\} | | |
| johnston-linear-matrix-algebra_FO1713 | 170 | 1.000 | \operatorname{range}\left(A^{T}\right):\{(1,0,0,-1),(0,1,1,0)\} | | |
| johnston-linear-matrix-algebra_FO1714 | 170 | 1.000 | x_{1}-x_{4}=0 | | |
| johnston-linear-matrix-algebra_FO1715 | 170 | 0.995 | \{(0,-1,1,0),(1,0,0,1)\} | | |
| johnston-linear-matrix-algebra_FO1716 | 171 | 1.000 | \operatorname{range}(A)=\mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1717 | 171 | 1.000 | \operatorname{null}(A)=\{\mathbf{0}\} | | |
| johnston-linear-matrix-algebra_FO1718 | 171 | 0.990 | \left(A^{T}\right)=\mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1719 | 171 | 0.990 | \left(A^{T}\right)=\{\mathbf{0}\} | | |
| johnston-linear-matrix-algebra_FO1720 | 171 | 0.531 | \operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO1721 | 171 | 1.000 | A=\mathbf{u u}^{T} | | |
| johnston-linear-matrix-algebra_FO1722 | 171 | 0.823 | \operatorname{rank}(A)=1 | | |
| johnston-linear-matrix-algebra_FO1723 | 171 | 0.406 | ((1,2,1),(1,-1,0)) | | |
| johnston-linear-matrix-algebra_FO1724 | 172 | 1.000 | \operatorname{rank}(A) \leq \min \{m, n\} | | |
| johnston-linear-matrix-algebra_FO1725 | 172 | 1.000 | \operatorname{rank}(A)=\min \{m, n\} | | |
| johnston-linear-matrix-algebra_FO1726 | 172 | 0.994 | \operatorname{rank}(A) \leq n | | |
| johnston-linear-matrix-algebra_FO1727 | 172 | 1.000 | \operatorname{rank}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1728 | 172 | 1.000 | C=\left[\begin{array}{ccccc}0 & 0 & -2 & 2 & -2 \\ 2 & -2 & -1 & 3 & 3 \\ -1 & 1 & -1 & 0 & -3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1729 | 173 | 1.000 | \operatorname{rank}(A)=n | | |
| johnston-linear-matrix-algebra_FO1730 | 173 | 1.000 | \operatorname{rank}(B)=2 | | |
| johnston-linear-matrix-algebra_FO1731 | 173 | 1.000 | \operatorname{rank}(C)=2 | | |
| johnston-linear-matrix-algebra_FO1732 | 173 | 1.000 | \operatorname{nullity}(A)=0 | | |
| johnston-linear-matrix-algebra_FO1733 | 173 | 1.000 | \operatorname{rank}(A)+\operatorname{nullity}(A)=n | | |
| johnston-linear-matrix-algebra_FO1734 | 173 | 1.000 | r=\operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO1735 | 173 | 1.000 | r | | |
| johnston-linear-matrix-algebra_FO1736 | 173 | 1.000 | n-r | | |
| johnston-linear-matrix-algebra_FO1737 | 173 | 0.996 | \operatorname{nullity}(A)=\operatorname{dim}(\operatorname{null}(A))=n-r | | |
| johnston-linear-matrix-algebra_FO1738 | 174 | 1.000 | \operatorname{rank}(A)=3 | | |
| johnston-linear-matrix-algebra_FO1739 | 174 | 1.000 | \operatorname{nullity}(A)=5-3=2 | | |
| johnston-linear-matrix-algebra_FO1740 | 174 | 1.000 | \mathbb{R}^{5} | | |
| johnston-linear-matrix-algebra_FO1741 | 175 | 1.000 | \operatorname{rank}(A B)=\operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO1742 | 175 | 1.000 | \sum_{j=1}^{n} | | |
| johnston-linear-matrix-algebra_FO1743 | 175 | 1.000 | \sum_{j=1}^{n} b_{j, 1} \mathbf{a}_{j}= | | |
| johnston-linear-matrix-algebra_FO1744 | 175 | 1.000 | b_{1,1} \mathbf{a}_{1}+\cdots+b_{n, 1} \mathbf{a}_{n} | | |
| johnston-linear-matrix-algebra_FO1746 | 175 | 1.000 | \operatorname{rank}(A+B)=\operatorname{rank}(A)+\operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1747 | 175 | 0.813 | \operatorname{rank}(A B)=\operatorname{rank}(A) \times \operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1748 | 175 | 0.994 | \operatorname{rank}(A+B) | | |
| johnston-linear-matrix-algebra_FO1749 | 175 | 0.994 | \operatorname{rank}(A B) | | |
| johnston-linear-matrix-algebra_FO1750 | 175 | 1.000 | \operatorname{rank}(A+B) \leq \operatorname{rank}(A)+\operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1751 | 175 | 1.000 | \operatorname{rank}(A B) \leq \min \{\operatorname{rank}(A), \operatorname{rank}(B)\} | | |
| johnston-linear-matrix-algebra_FO1752 | 175 | 0.996 | \mathbf{a}_{1}, \mathbf{a}_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO1753 | 175 | 0.996 | \mathbf{b}_{1}, \mathbf{b}_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO1754 | 175 | 1.000 | \mathbf{a}_{1}+\mathbf{b}_{1}, \mathbf{a}_{2}+\mathbf{b}_{2}, \ldots, \mathbf{a}_{n}+\mathbf{b}_{n} | | |
| johnston-linear-matrix-algebra_FO1755 | 175 | 0.999 | \mathbf{a}_{1}, \mathbf{b}_{1}, \mathbf{a}_{2}, \mathbf{b}_{2}, \ldots, \mathbf{a}_{n}, \mathbf{b}_{n} | | |
| johnston-linear-matrix-algebra_FO1756 | 175 | 0.955 | \operatorname{range}(A+B) | | |
| johnston-linear-matrix-algebra_FO1757 | 175 | 0.955 | \operatorname{rank}(A+ | | |
| johnston-linear-matrix-algebra_FO1758 | 175 | 0.863 | \operatorname{rank}(A)+\operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1759 | 176 | 1.000 | \operatorname{range}(A B) \subseteq \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1760 | 176 | 0.998 | \operatorname{rank}(A B) \leq \operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO1761 | 176 | 1.000 | \operatorname{rank}(A B) \leq \operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1762 | 176 | 0.999 | B=\{(1,1),(1,2),(3,4)\}, \mathcal{S}=\mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO1763 | 176 | 1.000 | B=\{(1,1),(1,-1)\}, \mathcal{S}=\mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO1764 | 176 | 0.963 | B=\{(1,1,1),(2,1,-1)\}, \mathcal{S}=\mathbb{R}^{3} | | |
| johnston-linear-matrix-algebra_FO1765 | 176 | 0.993 | B=\{(2,2,1),(0,1,1)\}, \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1766 | 176 | 1.000 | x+y-z=0 | | |
| johnston-linear-matrix-algebra_FO1767 | 176 | 0.997 | B=\{(1,0,1),(2,1,3)\}, \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO1768 | 176 | 1.000 | B=\{(1,3,2),(-1,-2,1),(1,4,5)\}, \mathcal{S}=\operatorname{span}(B) | | |
| johnston-linear-matrix-algebra_FO1769 | 176 | 0.958 | 2 x+y+z=0 | | |
| johnston-linear-matrix-algebra_FO1770 | 176 | 1.000 | z=4 x-3 y | | |
| johnston-linear-matrix-algebra_FO1771 | 176 | 1.000 | 2 x-y+2 z=0 | | |
| johnston-linear-matrix-algebra_FO1772 | 176 | 1.000 | \operatorname{span}(\{(1,3,2,1),(-1,-2,1,0),(1,4,-1,1)\}) | | |
| johnston-linear-matrix-algebra_FO1773 | 176 | 1.000 | \{(5,5,5,1),(3,6,3,2),(4,1,4,4),(1,4,1,4)\} | | |
| johnston-linear-matrix-algebra_FO1774 | 176 | 0.942 | \{(1,1,4,5,1),(2,-7,4,2,2),(5,2,4,5,3) | | |
| johnston-linear-matrix-algebra_FO1775 | 176 | 1.000 | \{(3,4,-1,1,3),(-1,1,3,0,1),(4,6,-5,3,6) | | |
| johnston-linear-matrix-algebra_FO1776 | 176 | 0.811 | \{(2,3,1,1),(3,3,2,3),(2,1,2,3)\} | | |
| johnston-linear-matrix-algebra_FO1777 | 176 | 0.993 | \{(3,2,0,2),(0,1,3,3),(2,1,1,1),(0,2,3,2)\} | | |
| johnston-linear-matrix-algebra_FO1778 | 176 | 1.000 | \{(2,4,3,4),(3,2,-1,-1)\} | | |
| johnston-linear-matrix-algebra_FO1779 | 176 | 1.000 | \{(-1,3,6,3),(1,3,5,0),(3,0,-1,-3)\} | | |
| johnston-linear-matrix-algebra_FO1780 | 176 | 1.000 | \operatorname{range}(A), \operatorname{null}(A), \operatorname{range}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1781 | 176 | 1.000 | \left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1782 | 176 | 1.000 | \left[\begin{array}{ccccc}1 & 2 & 0 & 3 & 0 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1783 | 176 | 1.000 | \left[\begin{array}{ccccc}0 & -4 & 0 & 2 & 1 \\ -1 & 2 & 1 & 2 & 1 \\ -2 & 0 & 2 & 6 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1784 | 176 | 1.000 | \left[\begin{array}{ccccc}2 & -1 & 2 & 3 & 5 \\ -1 & 1 & -2 & -2 & -2 \\ 1 & -1 & 2 & 1 & 2 \\ 1 & -2 & 4 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1785 | 176 | 0.651 | \left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1786 | 176 | 1.000 | \left[\begin{array}{cccc}5 & 4 & 8 & 1 \\ 3 & -2 & 7 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1787 | 176 | 1.000 | \left[\begin{array}{ll}1 & 0 \\ 0 & 2 \\ 3 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1788 | 176 | 1.000 | \left[\begin{array}{cccc}1 & 2 & 1 & 0 \\ 2 & 1 & 2 & -3 \\ 0 & 3 & 1 & 2 \\ 0 & 0 & 2 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1789 | 177 | 0.902 | A \in \mathcal{M}_{3,7} | | |
| johnston-linear-matrix-algebra_FO1790 | 177 | 0.902 | (A) \leq 3 | | |
| johnston-linear-matrix-algebra_FO1791 | 177 | 1.000 | \operatorname{rank}(A B)=\operatorname{rank}(A) \cdot \operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1792 | 177 | 1.000 | \operatorname{rank}(A B)=\min \{\operatorname{rank}(A), \operatorname{rank}(B)\} | | |
| johnston-linear-matrix-algebra_FO1793 | 177 | 1.000 | \operatorname{null}(A)= | | |
| johnston-linear-matrix-algebra_FO1794 | 177 | 1.000 | \operatorname{null}(B) | | |
| johnston-linear-matrix-algebra_FO1795 | 177 | 0.603 | \operatorname{nullity}(A)=\operatorname{nullity}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1796 | 177 | 1.000 | A \in \mathcal{M}_{4} | | |
| johnston-linear-matrix-algebra_FO1797 | 177 | 1.000 | A \in \mathcal{M}_{3} | | |
| johnston-linear-matrix-algebra_FO1798 | 177 | 0.715 | \operatorname{rank}\left(J_{n}\right) | | |
| johnston-linear-matrix-algebra_FO1799 | 177 | 1.000 | A \mathbf{v} \neq B \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO1800 | 177 | 1.000 | \mathbf{v} \neq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1801 | 177 | 0.884 | \operatorname{rank}\left(A_{n}\right)=2 | | |
| johnston-linear-matrix-algebra_FO1802 | 177 | 0.884 | n \geq 2 | | |
| johnston-linear-matrix-algebra_FO1803 | 177 | 0.814 | \left[\begin{array}{ll}1 & x \\ 2 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1804 | 177 | 1.000 | \left[\begin{array}{cc}x-1 & -1 \\ 1 & 1-x\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1805 | 177 | 1.000 | \left[\begin{array}{ccc}1 & 2 & x \\ 2 & 3-x & 3 x+1 \\ -x & -2 x & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1806 | 177 | 0.987 | \left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & x\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1807 | 177 | 0.931 | \mid \operatorname{rank}(A)- | | |
| johnston-linear-matrix-algebra_FO1808 | 177 | 0.967 | \operatorname{rank}(B) \mid \leq 1 | | |
| johnston-linear-matrix-algebra_FO1809 | 177 | 0.967 | \operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1810 | 177 | 1.000 | |\operatorname{rank}(A)-\operatorname{rank}(B)| \leq k | | |
| johnston-linear-matrix-algebra_FO1811 | 177 | 1.000 | x=1 | | |
| johnston-linear-matrix-algebra_FO1812 | 177 | 1.000 | x \neq 1 | | |
| johnston-linear-matrix-algebra_FO1813 | 177 | 1.000 | \operatorname{rank}(A B)=\operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO1814 | 177 | 1.000 | \operatorname{rank}(A B)= | | |
| johnston-linear-matrix-algebra_FO1815 | 177 | 1.000 | \operatorname{rank}(A) \leq n / 2 | | |
| johnston-linear-matrix-algebra_FO1816 | 178 | 1.000 | \mathcal{S}_{1} | | |
| johnston-linear-matrix-algebra_FO1817 | 178 | 1.000 | \mathcal{S}_{2} | | |
| johnston-linear-matrix-algebra_FO1818 | 178 | 1.000 | \mathcal{S}_{1} \subseteq \mathcal{S}_{2} | | |
| johnston-linear-matrix-algebra_FO1819 | 178 | 1.000 | \operatorname{dim}\left(\mathcal{S}_{1}\right) \leq \operatorname{dim}\left(\mathcal{S}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO1820 | 178 | 1.000 | \operatorname{dim}\left(\mathcal{S}_{1}\right)=\operatorname{dim}\left(\mathcal{S}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO1821 | 178 | 1.000 | \mathcal{S}_{1}=\mathcal{S}_{2} | | |
| johnston-linear-matrix-algebra_FO1822 | 178 | 0.893 | -1,2,3 | | |
| johnston-linear-matrix-algebra_FO1823 | 178 | 1.000 | \mathbf{v} \in \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1824 | 178 | 1.000 | \mathbf{w} \in \operatorname{null}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1825 | 178 | 0.999 | \mathbf{w} \in \operatorname{range}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1826 | 178 | 1.000 | \operatorname{null}(A)=\operatorname{null}(B) | | |
| johnston-linear-matrix-algebra_FO1827 | 178 | 1.000 | \operatorname{range}\left(A^{T}\right)=\operatorname{range}\left(B^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1828 | 178 | 0.981 | \operatorname{nullity}(A B) \leq \operatorname{nullity}(A)+\operatorname{nullity}(B) | | |
| johnston-linear-matrix-algebra_FO1829 | 178 | 1.000 | \operatorname{rank}(A B) \geq \operatorname{rank}(A)+\operatorname{rank}(B)-n | | |
| johnston-linear-matrix-algebra_FO1830 | 178 | 1.000 | A=\mathbf{v w}^{T} | | |
| johnston-linear-matrix-algebra_FO1831 | 178 | 1.000 | m \geq n | | |
| johnston-linear-matrix-algebra_FO1832 | 178 | 1.000 | B \in \mathcal{M}_{n, m} | | |
| johnston-linear-matrix-algebra_FO1833 | 178 | 1.000 | B A=I_{n} | | |
| johnston-linear-matrix-algebra_FO1834 | 178 | 0.999 | n \geq m | | |
| johnston-linear-matrix-algebra_FO1835 | 178 | 0.999 | \operatorname{rank}(A)=m | | |
| johnston-linear-matrix-algebra_FO1836 | 178 | 1.000 | C \in \mathcal{M}_{n, m} | | |
| johnston-linear-matrix-algebra_FO1837 | 178 | 1.000 | A C=I_{m} | | |
| johnston-linear-matrix-algebra_FO1838 | 178 | 1.000 | R \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO1839 | 178 | 1.000 | A^{-1} \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO1840 | 179 | 0.896 | (\mathbf{x}=\mathbf{0}) | | |
| johnston-linear-matrix-algebra_FO1841 | 179 | 1.000 | P \in \mathcal{M}_{m} | | |
| johnston-linear-matrix-algebra_FO1842 | 179 | 1.000 | A=P B | | |
| johnston-linear-matrix-algebra_FO1843 | 179 | 1.000 | A=P R | | |
| johnston-linear-matrix-algebra_FO1844 | 180 | 0.984 | A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1845 | 180 | 0.984 | B=\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1846 | 180 | 1.000 | A=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1847 | 180 | 1.000 | B=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1848 | 180 | 0.993 | A=\left[\begin{array}{ccc}1 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & -1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1849 | 180 | 0.993 | B=\left[\begin{array}{lll}3 & 3 & 2 \\ 0 & 0 & 0 \\ 0 & 3 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1850 | 180 | 1.000 | A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 1 & 0 & 2 \\ 2 & -1 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1851 | 180 | 1.000 | B=\left[\begin{array}{ccc}1 & 2 & -1 \\ 1 & 2 & 0 \\ 2 & 4 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1852 | 180 | 0.946 | A=\left[\begin{array}{ccc}1 & -1 & 0 \\ -1 & 1 & -3 \\ 1 & -2 & -1 \\ 1 & 1 & -3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1853 | 180 | 0.946 | B=\left[\begin{array}{ccc}2 & -1 & 3 \\ 0 & -1 & 2 \\ -1 & -1 & 1 \\ -1 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1854 | 180 | 1.000 | A=\left[\begin{array}{llll}1 & 1 & 4 & 3 \\ 1 & 2 & 3 & 1 \\ 3 & 3 & 3 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1855 | 180 | 1.000 | B=\left[\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 2 \\ 4 & 2 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO1856 | 180 | 1.000 | R \mathbf{x}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1857 | 180 | 1.000 | A \mathbf{v}_{j}=B \mathbf{v}_{j} | | |
| johnston-linear-matrix-algebra_FO1858 | 180 | 0.961 | 4 \times 4 | | |
| johnston-linear-matrix-algebra_FO1859 | 180 | 0.867 | A=B C | | |
| johnston-linear-matrix-algebra_FO1860 | 180 | 1.000 | \operatorname{null}(A)=\operatorname{null}\left(A^{T} A\right) | | |
| johnston-linear-matrix-algebra_FO1861 | 180 | 1.000 | \mathbf{x} \in \operatorname{null}\left(A^{T} A\right) | | |
| johnston-linear-matrix-algebra_FO1862 | 180 | 1.000 | \|A \mathbf{x}\|^{2} | | |
| johnston-linear-matrix-algebra_FO1863 | 180 | 1.000 | \operatorname{rank}\left(A^{T} A\right)=\operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO1864 | 180 | 0.975 | \operatorname{range}(A)=\operatorname{range}\left(A A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1865 | 180 | 0.998 | \operatorname{rank}\left(A A^{T}\right)=\operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO1866 | 180 | 0.994 | \operatorname{range}(A)=\operatorname{range}\left(A^{T} A\right) | | |
| johnston-linear-matrix-algebra_FO1867 | 180 | 1.000 | \operatorname{null}(A)=\operatorname{null}\left(A A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO1868 | 181 | 1.000 | a(b+c)=a b+a c | | |
| johnston-linear-matrix-algebra_FO1869 | 181 | 1.000 | a+i b | | |
| johnston-linear-matrix-algebra_FO1870 | 181 | 1.000 | i^{2}=-1 | | |
| johnston-linear-matrix-algebra_FO1871 | 181 | 0.996 | \mathbb{Z}_{2} | | |
| johnston-linear-matrix-algebra_FO1872 | 181 | 1.000 | \{0,1\} | | |
| johnston-linear-matrix-algebra_FO1873 | 181 | 0.978 | 1+1=0 | | |
| johnston-linear-matrix-algebra_FO1874 | 181 | 0.978 | 1+1=2 | | |
| johnston-linear-matrix-algebra_FO1875 | 181 | 1.000 | \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1876 | 182 | 0.996 | 1=0-1 | | |
| johnston-linear-matrix-algebra_FO1877 | 182 | 1.000 | \mathbb{Z}_{2}^{n} | | |
| johnston-linear-matrix-algebra_FO1878 | 182 | 1.000 | \left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)= | | |
| johnston-linear-matrix-algebra_FO1879 | 182 | 1.000 | 2^{n} | | |
| johnston-linear-matrix-algebra_FO1880 | 183 | 1.000 | 0 R_{j} | | |
| johnston-linear-matrix-algebra_FO1881 | 183 | 1.000 | 1 R_{j} | | |
| johnston-linear-matrix-algebra_FO1882 | 183 | 0.980 | x_{5}=1, x_{4}=1, x_{2}=1-x_{3} | | |
| johnston-linear-matrix-algebra_FO1883 | 183 | 0.980 | x_{1}=x_{3} | | |
| johnston-linear-matrix-algebra_FO1884 | 183 | 0.980 | x_{3}=-x_{3} | | |
| johnston-linear-matrix-algebra_FO1885 | 183 | 1.000 | x_{3}=0 | | |
| johnston-linear-matrix-algebra_FO1886 | 183 | 1.000 | x_{3}=1 | | |
| johnston-linear-matrix-algebra_FO1887 | 183 | 1.000 | 2^{k} | | |
| johnston-linear-matrix-algebra_FO1889 | 184 | 1.000 | \mathbf{v}_{\mathrm{s}}, \mathbf{v}_{\mathrm{e}} \in \mathbb{Z}_{2}^{16} | | |
| johnston-linear-matrix-algebra_FO1890 | 184 | 0.994 | \mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{16} | | |
| johnston-linear-matrix-algebra_FO1891 | 184 | 0.994 | \mathbf{v}_{\mathrm{s}} | | |
| johnston-linear-matrix-algebra_FO1892 | 184 | 1.000 | \mathbf{x} \in \mathbb{Z}_{2}^{16} | | |
| johnston-linear-matrix-algebra_FO1893 | 184 | 1.000 | x_{j}=1 | | |
| johnston-linear-matrix-algebra_FO1894 | 185 | 0.937 | x_{j}=0 | | |
| johnston-linear-matrix-algebra_FO1895 | 185 | 0.937 | x_{1}, x_{2}, \ldots, x_{16} | | |
| johnston-linear-matrix-algebra_FO1896 | 185 | 0.919 | A \mathbf{x}=\mathbf{v}_{\mathrm{e}}-\mathbf{v}_{\mathrm{s}} | | |
| johnston-linear-matrix-algebra_FO1897 | 185 | 0.919 | A=\left[\mathbf{a}_{1}\left|\mathbf{a}_{2}\right| \cdots \mid \mathbf{a}_{16}\right] | | |
| johnston-linear-matrix-algebra_FO1898 | 185 | 0.973 | \mathbf{v}_{\mathrm{e}}-\mathbf{v}_{\mathrm{s}} \in \mathbb{Z}_{2}^{16} | | |
| johnston-linear-matrix-algebra_FO1899 | 186 | 0.999 | \left[A \mid \mathbf{v}_{\mathrm{e}}-\mathbf{v}_{\mathrm{s}}\right] | | |
| johnston-linear-matrix-algebra_FO1900 | 186 | 1.000 | 2^{4}=16 | | |
| johnston-linear-matrix-algebra_FO1901 | 186 | 1.000 | x_{13}, x_{14}, x_{15} | | |
| johnston-linear-matrix-algebra_FO1902 | 186 | 1.000 | x_{16} | | |
| johnston-linear-matrix-algebra_FO1903 | 186 | 0.992 | x_{13}=0, x_{14}=1, x_{15}=0, x_{16}=0 | | |
| johnston-linear-matrix-algebra_FO1904 | 187 | 0.992 | 2^{16}=65536 | | |
| johnston-linear-matrix-algebra_FO1905 | 187 | 1.000 | 65536 / 16=4096 | | |
| johnston-linear-matrix-algebra_FO1906 | 188 | 0.994 | \mathbf{v}_{\mathrm{s}}=(0,0,0,0,1,1,1,1,0) | | |
| johnston-linear-matrix-algebra_FO1907 | 188 | 0.994 | \mathbf{v}_{\mathrm{e}}=(1,1,1,1,1,1,1,1,1) | | |
| johnston-linear-matrix-algebra_FO1908 | 188 | 0.916 | 9 \times 9 | | |
| johnston-linear-matrix-algebra_FO1909 | 188 | 0.916 | 1 \leq j \leq 9 | | |
| johnston-linear-matrix-algebra_FO1910 | 188 | 1.000 | [I \mid \mathbf{x}] | | |
| johnston-linear-matrix-algebra_FO1911 | 188 | 1.000 | \mathbf{x}=(1,1,0,0,0,0,1,1,0) | | |
| johnston-linear-matrix-algebra_FO1912 | 189 | 1.000 | \mathbb{Z}_{2}^{k} | | |
| johnston-linear-matrix-algebra_FO1913 | 189 | 1.000 | A \mathbf{x}=\mathbf{1} | | |
| johnston-linear-matrix-algebra_FO1914 | 189 | 0.516 | \mathbf{1}=(1,1, \ldots, 1) \in \mathbb{Z}_{2}^{n} | | |
| johnston-linear-matrix-algebra_FO1915 | 189 | 1.000 | \mathbf{1} \in \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO1916 | 189 | 1.000 | A^{T}=A | | |
| johnston-linear-matrix-algebra_FO1917 | 189 | 1.000 | \mathbf{y} \in \operatorname{null}(A) | | |
| johnston-linear-matrix-algebra_FO1918 | 189 | 1.000 | \mathbf{y} \in \mathbb{Z}_{2}^{n} | | |
| johnston-linear-matrix-algebra_FO1919 | 189 | 1.000 | A \mathbf{y}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1920 | 189 | 1.000 | y_{j}=0 | | |
| johnston-linear-matrix-algebra_FO1921 | 189 | 0.808 | B^{T}=B | | |
| johnston-linear-matrix-algebra_FO1922 | 189 | 1.000 | B \mathbf{1}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1923 | 189 | 0.839 | B 1 | | |
| johnston-linear-matrix-algebra_FO1924 | 190 | 0.874 | \mathbf{1} | | |
| johnston-linear-matrix-algebra_FO1925 | 190 | 0.982 | \mathbb{Z}_{3} | | |
| johnston-linear-matrix-algebra_FO1926 | 190 | 0.982 | \{0,1,2\} | | |
| johnston-linear-matrix-algebra_FO1927 | 190 | 0.999 | \bmod 3 | | |
| johnston-linear-matrix-algebra_FO1928 | 190 | 1.000 | 2+2=4 | | |
| johnston-linear-matrix-algebra_FO1929 | 190 | 1.000 | 2+2=1 | | |
| johnston-linear-matrix-algebra_FO1930 | 190 | 0.919 | 4: 00 | | |
| johnston-linear-matrix-algebra_FO1931 | 190 | 0.919 | 9+7=16=12+4 | | |
| johnston-linear-matrix-algebra_FO1932 | 190 | 0.924 | \bmod -12 | | |
| johnston-linear-matrix-algebra_FO1933 | 190 | 0.924 | 9+7=4 | | |
| johnston-linear-matrix-algebra_FO1934 | 191 | 1.000 | 0-1=2,0-2=1 | | |
| johnston-linear-matrix-algebra_FO1935 | 191 | 1.000 | 1-2=2 | | |
| johnston-linear-matrix-algebra_FO1936 | 191 | 1.000 | \mathbb{Z}_{p} | | |
| johnston-linear-matrix-algebra_FO1937 | 191 | 1.000 | 0,1,2, \ldots, p-1 | | |
| johnston-linear-matrix-algebra_FO1938 | 191 | 1.000 | (w, x, y, z)= | | |
| johnston-linear-matrix-algebra_FO1939 | 192 | 1.000 | 3^{4}=81 | | |
| johnston-linear-matrix-algebra_FO1940 | 192 | 1.000 | \mathbb{Z}_{3}^{4} | | |
| johnston-linear-matrix-algebra_FO1941 | 192 | 1.000 | \mathbf{v}=(c, p, d, n) | | |
| johnston-linear-matrix-algebra_FO1942 | 192 | 1.000 | c, p, d | | |
| johnston-linear-matrix-algebra_FO1944 | 193 | 1.000 | \mathbf{v}=(c, p, d, n)=(0,2,1,0) | | |
| johnston-linear-matrix-algebra_FO1945 | 193 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2} | | |
| johnston-linear-matrix-algebra_FO1946 | 193 | 1.000 | \mathbf{v}_{1}+\mathbf{v}_{2}+\mathbf{v}_{3}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1947 | 193 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2} \in \mathbb{Z}_{3}^{4} | | |
| johnston-linear-matrix-algebra_FO1948 | 193 | 1.000 | \mathbf{v}_{3} \in \mathbb{Z}_{3}^{4} | | |
| johnston-linear-matrix-algebra_FO1949 | 193 | 1.000 | \mathbf{v}_{3}=-\mathbf{v}_{1}-\mathbf{v}_{2} | | |
| johnston-linear-matrix-algebra_FO1950 | 193 | 1.000 | 26 \times 3=78 | | |
| johnston-linear-matrix-algebra_FO1951 | 193 | 1.000 | \mathbf{v}_{1}-\mathbf{v}_{2}=\mathbf{v}_{2}-\mathbf{v}_{3} | | |
| johnston-linear-matrix-algebra_FO1952 | 193 | 0.999 | (x, 2, y, x) | | |
| johnston-linear-matrix-algebra_FO1953 | 194 | 1.000 | \mathbb{Z}_{5} | | |
| johnston-linear-matrix-algebra_FO1954 | 194 | 1.000 | 2 x=1 | | |
| johnston-linear-matrix-algebra_FO1955 | 194 | 0.982 | \{0,1,2, \ldots, p-1\} | | |
| johnston-linear-matrix-algebra_FO1956 | 194 | 1.000 | y \in \mathbb{Z}_{p} | | |
| johnston-linear-matrix-algebra_FO1957 | 194 | 1.000 | x \in \mathbb{Z}_{p} | | |
| johnston-linear-matrix-algebra_FO1958 | 194 | 1.000 | x y=1 | | |
| johnston-linear-matrix-algebra_FO1959 | 194 | 1.000 | 2 x=3 | | |
| johnston-linear-matrix-algebra_FO1960 | 195 | 0.982 | 2 \cdot 3=1 | | |
| johnston-linear-matrix-algebra_FO1961 | 195 | 1.000 | x=4 | | |
| johnston-linear-matrix-algebra_FO1962 | 195 | 0.998 | w, x | | |
| johnston-linear-matrix-algebra_FO1963 | 195 | 1.000 | z=2, x=1-y | | |
| johnston-linear-matrix-algebra_FO1964 | 195 | 1.000 | w=3-2 y | | |
| johnston-linear-matrix-algebra_FO1965 | 195 | 1.000 | p=2 | | |
| johnston-linear-matrix-algebra_FO1966 | 195 | 1.000 | p^{k} | | |
| johnston-linear-matrix-algebra_FO1967 | 196 | 0.863 | v, w, x, y, z | | |
| johnston-linear-matrix-algebra_FO1968 | 197 | 0.999 | k \times k | | |
| johnston-linear-matrix-algebra_FO1969 | 197 | 0.988 | x_{1}, x_{2}, \ldots, x_{n} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO1970 | 198 | 1.000 | \mathbf{x} \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO1971 | 198 | 0.781 | A \mathbf{x} \leq \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1972 | 198 | 1.000 | A \in \mathcal{M}_{m, n}, \mathbf{b} \in \mathbb{R}^{m}, \mathbf{c} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO1973 | 198 | 0.988 | \mathbf{c} \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO1974 | 198 | 1.000 | -(\mathbf{c} \cdot \mathbf{x}) | | |
| johnston-linear-matrix-algebra_FO1975 | 199 | 0.699 | A, \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO1976 | 199 | 0.699 | \mathbf{c} | | |
| johnston-linear-matrix-algebra_FO1977 | 199 | 1.000 | \left(x_{1}, x_{2}\right)=(1,2) | | |
| johnston-linear-matrix-algebra_FO1978 | 199 | 1.000 | -x_{1}-2 x_{2}=-5 | | |
| johnston-linear-matrix-algebra_FO1979 | 199 | 1.000 | x_{1}+2 x_{2}=5 | | |
| johnston-linear-matrix-algebra_FO1980 | 199 | 1.000 | \geq | | |
| johnston-linear-matrix-algebra_FO1981 | 199 | 1.000 | \leq | | |
| johnston-linear-matrix-algebra_FO1982 | 199 | 1.000 | x=a | | |
| johnston-linear-matrix-algebra_FO1983 | 199 | 1.000 | x \leq a | | |
| johnston-linear-matrix-algebra_FO1984 | 199 | 1.000 | x \geq a | | |
| johnston-linear-matrix-algebra_FO1985 | 199 | 1.000 | -x \leq-a | | |
| johnston-linear-matrix-algebra_FO1986 | 199 | 0.936 | 3 x_{1}-4 x_{2} \geq 1 | | |
| johnston-linear-matrix-algebra_FO1987 | 199 | 1.000 | -3 x_{1}+4 x_{2} \leq-1 | | |
| johnston-linear-matrix-algebra_FO1988 | 199 | 1.000 | 2 x_{1}+ | | |
| johnston-linear-matrix-algebra_FO1989 | 199 | 1.000 | x_{2}=5 | | |
| johnston-linear-matrix-algebra_FO1990 | 199 | 1.000 | 2 x_{1}+x_{2} \leq 5 | | |
| johnston-linear-matrix-algebra_FO1991 | 199 | 1.000 | -2 x_{1}-x_{2} \leq-5 | | |
| johnston-linear-matrix-algebra_FO1992 | 200 | 0.900 | x_{1}+3\left(x_{2}^{+}-x_{2}^{-}\right) \leq 4 | | |
| johnston-linear-matrix-algebra_FO1993 | 200 | 1.000 | x_{1}+3 x_{2}^{+}-3 x_{2}^{-} \leq 4 | | |
| johnston-linear-matrix-algebra_FO1994 | 200 | 1.000 | x=x^{+}-x^{-} | | |
| johnston-linear-matrix-algebra_FO1995 | 200 | 1.000 | x^{+} \geq 0 | | |
| johnston-linear-matrix-algebra_FO1996 | 200 | 1.000 | x^{-} \geq 0 | | |
| johnston-linear-matrix-algebra_FO1997 | 200 | 1.000 | x_{2} \geq 0 | | |
| johnston-linear-matrix-algebra_FO1998 | 200 | 1.000 | x_{2}=x_{2}^{+}-x_{2}^{-} | | |
| johnston-linear-matrix-algebra_FO1999 | 200 | 1.000 | x_{2}^{+}, x_{2}^{-} \geq 0 | | |
| johnston-linear-matrix-algebra_FO2000 | 200 | 1.000 | x_{2}^{+}-x_{2}^{-} | | |
| johnston-linear-matrix-algebra_FO2001 | 201 | 0.989 | x_{1}^{+}-x_{1}^{-} | | |
| johnston-linear-matrix-algebra_FO2002 | 201 | 0.989 | x_{1}^{+}, x_{1}^{-} \geq 0 | | |
| johnston-linear-matrix-algebra_FO2003 | 201 | 1.000 | \mathbf{x}=\left(x_{1}, x_{2}\right) | | |
| johnston-linear-matrix-algebra_FO2004 | 201 | 1.000 | x_{1} \geq 0 | | |
| johnston-linear-matrix-algebra_FO2005 | 201 | 1.000 | x_{2} \leq-x_{1}+3 | | |
| johnston-linear-matrix-algebra_FO2006 | 201 | 1.000 | x_{2}=-x_{1}+3 | | |
| johnston-linear-matrix-algebra_FO2007 | 202 | 1.000 | f\left(x_{1}, x_{2}\right)=x_{1}+2 x_{2} | | |
| johnston-linear-matrix-algebra_FO2008 | 202 | 1.000 | x_{2} \leq x_{1}+1 | | |
| johnston-linear-matrix-algebra_FO2009 | 202 | 1.000 | x_{2}=x_{1}+1 | | |
| johnston-linear-matrix-algebra_FO2010 | 202 | 1.000 | z=x_{1}+2 x_{2} | | |
| johnston-linear-matrix-algebra_FO2011 | 202 | 1.000 | x_{2}=-x_{1} / 2+z / 2 | | |
| johnston-linear-matrix-algebra_FO2012 | 202 | 1.000 | -1 / 2 | | |
| johnston-linear-matrix-algebra_FO2013 | 202 | 1.000 | z / 2 | | |
| johnston-linear-matrix-algebra_FO2014 | 202 | 1.000 | z=5 | | |
| johnston-linear-matrix-algebra_FO2015 | 202 | 1.000 | \mathbf{x}=\left(x_{1}, x_{2}\right)=(1,2) | | |
| johnston-linear-matrix-algebra_FO2016 | 203 | 0.999 | x_{1}=x_{2}=0 | | |
| johnston-linear-matrix-algebra_FO2017 | 203 | 0.980 | \left(x_{1}, x_{2}\right) | | |
| johnston-linear-matrix-algebra_FO2018 | 203 | 1.000 | x_{1}+2 x_{2}=x_{1}+2\left(x_{1}+1\right)=3 x_{1}+2 | | |
| johnston-linear-matrix-algebra_FO2019 | 203 | 1.000 | \left(x_{1}-x_{2}\right)+\left(-x_{1}+x_{2}\right)=0 \leq-1-1=-2 | | |
| johnston-linear-matrix-algebra_FO2020 | 203 | 1.000 | \infty | | |
| johnston-linear-matrix-algebra_FO2021 | 203 | 1.000 | -\infty | | |
| johnston-linear-matrix-algebra_FO2022 | 204 | 1.000 | x_{2}-x_{3} | | |
| johnston-linear-matrix-algebra_FO2023 | 204 | 1.000 | x_{2}, x_{3} \geq 0 | | |
| johnston-linear-matrix-algebra_FO2024 | 204 | 1.000 | \mathbf{b} \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2025 | 204 | 1.000 | \mathbf{b}=(1,2,3) \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2026 | 204 | 1.000 | x \leq 5 | | |
| johnston-linear-matrix-algebra_FO2027 | 204 | 1.000 | s+x=5, s \geq 0 | | |
| johnston-linear-matrix-algebra_FO2028 | 204 | 1.000 | s | | |
| johnston-linear-matrix-algebra_FO2029 | 204 | 0.997 | z=\mathbf{c} \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO2030 | 204 | 1.000 | z-\mathbf{c} \cdot \mathbf{x}=0 | | |
| johnston-linear-matrix-algebra_FO2031 | 205 | 1.000 | z-\mathbf{c}^{T} \mathbf{x}=0 | | |
| johnston-linear-matrix-algebra_FO2032 | 205 | 1.000 | [\mathbf{0}|I| A \mid \mathbf{b}] | | |
| johnston-linear-matrix-algebra_FO2033 | 205 | 1.000 | z, s_{1}, s_{2} | | |
| johnston-linear-matrix-algebra_FO2034 | 205 | 1.000 | s_{3} | | |
| johnston-linear-matrix-algebra_FO2035 | 206 | 1.000 | z=0 | | |
| johnston-linear-matrix-algebra_FO2036 | 206 | 1.000 | z=2 x_{1}+x_{2}-x_{3} | | |
| johnston-linear-matrix-algebra_FO2037 | 206 | 1.000 | x_{2}=x_{3}=0 | | |
| johnston-linear-matrix-algebra_FO2038 | 206 | 1.000 | x_{1}=2 | | |
| johnston-linear-matrix-algebra_FO2039 | 206 | 0.990 | \begin{array}{lllllll}z & s_{1} & s_{2} & s_{3} & x_{1} & x_{2} & x_{3}\end{array} | | |
| johnston-linear-matrix-algebra_FO2042 | 206 | 1.000 | s_{1}, s_{3} | | |
| johnston-linear-matrix-algebra_FO2043 | 206 | 1.000 | s_{2}, x_{2} | | |
| johnston-linear-matrix-algebra_FO2044 | 206 | 1.000 | z=4 | | |
| johnston-linear-matrix-algebra_FO2045 | 207 | 1.000 | z=23 / 2-(7 / 2) s_{2}-(3 / 2) s_{3} | | |
| johnston-linear-matrix-algebra_FO2046 | 207 | 1.000 | s_{2}, s_{3} \geq 0 | | |
| johnston-linear-matrix-algebra_FO2047 | 207 | 1.000 | z=23 / 2 | | |
| johnston-linear-matrix-algebra_FO2048 | 207 | 0.636 | x_{1}=9 / 2 | | |
| johnston-linear-matrix-algebra_FO2049 | 207 | 0.636 | x_{2}=5 / 2 | | |
| johnston-linear-matrix-algebra_FO2050 | 207 | 0.636 | s_{1}=39 / 2 | | |
| johnston-linear-matrix-algebra_FO2051 | 207 | 0.636 | s_{2}=s_{3}=x_{3}=0 | | |
| johnston-linear-matrix-algebra_FO2052 | 207 | 1.000 | \left(x_{1}, x_{2}, x_{3}\right)=(0,0,0) | | |
| johnston-linear-matrix-algebra_FO2053 | 207 | 1.000 | \left(x_{1}, x_{2}, x_{3}\right)=(2,0,0) | | |
| johnston-linear-matrix-algebra_FO2054 | 207 | 0.755 | (9 / 2,5 / 2,0) | | |
| johnston-linear-matrix-algebra_FO2055 | 208 | 1.000 | \binom{m}{n} | | |
| johnston-linear-matrix-algebra_FO2059 | 208 | 1.000 | x_{\mathrm{p}}, x_{\mathrm{c}} | | |
| johnston-linear-matrix-algebra_FO2060 | 208 | 1.000 | x_{\mathrm{pc}} | | |
| johnston-linear-matrix-algebra_FO2061 | 209 | 1.000 | x_{\mathrm{p}}, x_{\mathrm{c}}, x_{\mathrm{pc}} \geq 0 | | |
| johnston-linear-matrix-algebra_FO2071 | 210 | 0.994 | \$ 151 | | |
| johnston-linear-matrix-algebra_FO2072 | 210 | 1.000 | x_{\mathrm{p}}=50 | | |
| johnston-linear-matrix-algebra_FO2073 | 210 | 1.000 | x_{\mathrm{c}}=70 | | |
| johnston-linear-matrix-algebra_FO2074 | 210 | 1.000 | x_{\mathrm{pc}}=30 | | |
| johnston-linear-matrix-algebra_FO2075 | 211 | 0.703 | \left(x_{1}, x_{2}\right)=(0.8,1.6) | | |
| johnston-linear-matrix-algebra_FO2076 | 211 | 1.000 | \left(x_{1}, x_{2}\right)=(0,1) | | |
| johnston-linear-matrix-algebra_FO2077 | 211 | 1.000 | -2 x_{1}+x_{2} \leq 0 | | |
| johnston-linear-matrix-algebra_FO2078 | 211 | 1.000 | x_{1}=0.8 | | |
| johnston-linear-matrix-algebra_FO2079 | 211 | 1.000 | x_{2}=1.6 | | |
| johnston-linear-matrix-algebra_FO2080 | 211 | 1.000 | \left(x_{1}, x_{2}\right)=(1,1) | | |
| johnston-linear-matrix-algebra_FO2081 | 211 | 1.000 | x_{1}+3 x_{2}=4 | | |
| johnston-linear-matrix-algebra_FO2082 | 211 | 0.997 | \mathbf{b} \not \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2083 | 212 | 1.000 | -y | | |
| johnston-linear-matrix-algebra_FO2084 | 212 | 1.000 | y \geq 0 | | |
| johnston-linear-matrix-algebra_FO2085 | 212 | 1.000 | y=2 | | |
| johnston-linear-matrix-algebra_FO2086 | 212 | 1.000 | \left(x_{1}, x_{2}, y\right)=\left(x_{1}, x_{2}, 0\right) | | |
| johnston-linear-matrix-algebra_FO2087 | 212 | 1.000 | \left(x_{1}, x_{2}, y\right)=(0,0,2) | | |
| johnston-linear-matrix-algebra_FO2088 | 212 | 1.000 | -2 x_{1}+x_{2}-y \leq-2 | | |
| johnston-linear-matrix-algebra_FO2093 | 213 | 0.982 | L P | | |
| johnston-linear-matrix-algebra_FO2094 | 213 | 1.000 | x_{1}=1 | | |
| johnston-linear-matrix-algebra_FO2095 | 213 | 1.000 | x_{2}=0 | | |
| johnston-linear-matrix-algebra_FO2096 | 213 | 1.000 | \left(x_{1}, x_{2}\right)=(1,0) | | |
| johnston-linear-matrix-algebra_FO2097 | 213 | 1.000 | -y<0 | | |
| johnston-linear-matrix-algebra_FO2103 | 214 | 1.000 | \left(x_{1}, x_{2}\right)=(4 / 3,2 / 3) | | |
| johnston-linear-matrix-algebra_FO2104 | 215 | 1.000 | y_{1}, y_{2} \geq 0 | | |
| johnston-linear-matrix-algebra_FO2105 | 215 | 0.998 | \left(x_{1}+x_{2} \leq 3\right) | | |
| johnston-linear-matrix-algebra_FO2106 | 215 | 1.000 | y_{1} | | |
| johnston-linear-matrix-algebra_FO2107 | 215 | 1.000 | y_{2} | | |
| johnston-linear-matrix-algebra_FO2108 | 215 | 1.000 | 3 y_{1}+y_{2} | | |
| johnston-linear-matrix-algebra_FO2109 | 215 | 1.000 | x_{1}+2 x_{2} | | |
| johnston-linear-matrix-algebra_FO2110 | 215 | 1.000 | y_{1}=3 / 2 | | |
| johnston-linear-matrix-algebra_FO2111 | 215 | 1.000 | y_{2}=1 / 2 | | |
| johnston-linear-matrix-algebra_FO2112 | 215 | 1.000 | x_{1}+2 x_{2} \leq 3(3 / 2)+(1 / 2)=5 | | |
| johnston-linear-matrix-algebra_FO2113 | 215 | 0.997 | A \in \mathcal{M}_{m, n}(\mathbb{R}), \mathbf{b} \in \mathbb{R}^{m}, \mathbf{c} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2114 | 215 | 0.997 | \mathbf{x} \in \mathbb{R}^{n}, \mathbf{y} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO2115 | 216 | 1.000 | \mathbf{y} \cdot(A \mathbf{x})=\mathbf{y}^{T} A \mathbf{x}= | | |
| johnston-linear-matrix-algebra_FO2116 | 216 | 1.000 | \left(A^{T} \mathbf{y}\right) \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO2117 | 216 | 1.000 | \mathbf{y} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO2118 | 216 | 1.000 | \mathbf{y} \cdot(A \mathbf{x}) \leq \mathbf{y} \cdot \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO2119 | 216 | 1.000 | A^{T} \mathbf{y} \geq \mathbf{c} | | |
| johnston-linear-matrix-algebra_FO2120 | 216 | 1.000 | \mathbf{y} \cdot(A \mathbf{x})=\left(A^{T} \mathbf{y}\right) \cdot \mathbf{x} \geq \mathbf{c} \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO2121 | 216 | 1.000 | \mathbf{c} \cdot \mathbf{x} \leq \mathbf{y} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2122 | 216 | 1.000 | \mathbf{c} \cdot \mathbf{x}=\mathbf{b} \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2123 | 216 | 0.984 | \mathbf{c} \cdot \mathbf{x}=5 | | |
| johnston-linear-matrix-algebra_FO2124 | 217 | 1.000 | \left(y_{1}, y_{2}\right)=(3 / 2,1 / 2) | | |
| johnston-linear-matrix-algebra_FO2125 | 217 | 0.679 | \mathbf{b} \cdot \mathbf{y}=5 | | |
| johnston-linear-matrix-algebra_FO2126 | 217 | 1.000 | \left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=(1,1,1,1,1) | | |
| johnston-linear-matrix-algebra_FO2127 | 217 | 0.998 | x_{2}+x_{3} | | |
| johnston-linear-matrix-algebra_FO2128 | 217 | 0.998 | x_{4}+x_{5} | | |
| johnston-linear-matrix-algebra_FO2129 | 218 | 1.000 | \mathbf{x}_{*} | | |
| johnston-linear-matrix-algebra_FO2130 | 218 | 1.000 | \mathbf{y}_{*} | | |
| johnston-linear-matrix-algebra_FO2131 | 218 | 1.000 | \left(y_{1}, y_{2}, y_{3}, y_{4}, y_{5}\right)=(1,1,1,1,1) / 3 | | |
| johnston-linear-matrix-algebra_FO2132 | 218 | 1.000 | \mathbf{x}_{*} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2133 | 218 | 1.000 | \mathbf{c} \cdot \mathbf{x}_{*} \geq \mathbf{c} \cdot \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO2134 | 218 | 1.000 | \mathbf{y}_{*} \in \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO2135 | 218 | 0.993 | \mathbf{b} \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2141 | 219 | 1.000 | -y \geq 1 | | |
| johnston-linear-matrix-algebra_FO2148 | 219 | 1.000 | 0 \geq 3 | | |
| johnston-linear-matrix-algebra_FO2149 | 220 | 1.000 | y_{1}-y_{2} | | |
| johnston-linear-matrix-algebra_FO2150 | 220 | 1.000 | y_{*} | | |
| johnston-linear-matrix-algebra_FO2151 | 220 | 0.852 | \geq 0 | | |
| johnston-linear-matrix-algebra_FO2156 | 221 | 1.000 | \left(x_{1}, x_{2}\right)=(0,-3) | | |
| johnston-linear-matrix-algebra_FO2157 | 221 | 0.999 | \mathbf{c} \cdot \mathbf{x}=-3 | | |
| johnston-linear-matrix-algebra_FO2158 | 221 | 0.965 | \left(y_{1}, y_{2}\right)=(1,0) | | |
| johnston-linear-matrix-algebra_FO2159 | 221 | 0.965 | \mathbf{b} \cdot \mathbf{y}=-3 | | |
| johnston-linear-matrix-algebra_FO2160 | 222 | 1.000 | \mathbf{x} \leq A^{-1} \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO2161 | 222 | 1.000 | \mathbf{c}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2162 | 222 | 1.000 | \mathbf{x} \geq \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2163 | 222 | 1.000 | \mathbf{c} \cdot \mathbf{x} \geq \mathbf{c} \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2164 | 222 | 1.000 | \mathbf{c}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}, \mathbf{c} \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2165 | 223 | 1.000 | 0<c \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2166 | 223 | 1.000 | \mathbf{x}_{1}, \ldots, \mathbf{x}_{k}, \mathbf{y} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2167 | 223 | 1.000 | \mathbf{y} \in | | |
| johnston-linear-matrix-algebra_FO2168 | 223 | 1.000 | \operatorname{span}\left(\mathbf{x}_{1}, \ldots, \mathbf{x}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO2169 | 223 | 0.827 | \mathbf{y} \in \operatorname{span}\left(\mathbf{x}_{1}, \ldots, \mathbf{x}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO2170 | 223 | 1.000 | \mathbf{x}_{1}, \ldots, \mathbf{x}_{k} | | |
| johnston-linear-matrix-algebra_FO2171 | 223 | 1.000 | \mathbf{x}_{1} \cdot \mathbf{z}=\cdots=\mathbf{x}_{k} \cdot \mathbf{z}=0 | | |
| johnston-linear-matrix-algebra_FO2172 | 223 | 1.000 | \mathbf{y} \cdot \mathbf{z}=0 | | |
| johnston-linear-matrix-algebra_FO2173 | 223 | 1.000 | \|A \mathbf{x}-\mathbf{b}\|_{1} | | |
| johnston-linear-matrix-algebra_FO2174 | 223 | 1.000 | |a| \leq b | | |
| johnston-linear-matrix-algebra_FO2175 | 223 | 1.000 | -b \leq a \leq b | | |
| johnston-linear-matrix-algebra_FO2176 | 223 | 1.000 | \|A \mathbf{x}-\mathbf{b}\|_{\infty} | | |
| johnston-linear-matrix-algebra_FO2177 | 223 | 1.000 | a_{1, j}+a_{2, j}+\cdots+a_{n, j}=1 | | |
| johnston-linear-matrix-algebra_FO2178 | 223 | 1.000 | A \mathbf{x}=\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO2179 | 223 | 1.000 | A^{T} \mathbf{y}=\mathbf{y} \ldots | | |
| johnston-linear-matrix-algebra_FO2180 | 223 | 1.000 | A=C R | | |
| johnston-linear-matrix-algebra_FO2181 | 223 | 1.000 | C \in \mathcal{M}_{m, r} | | |
| johnston-linear-matrix-algebra_FO2182 | 223 | 1.000 | R \in \mathcal{M}_{r, n} | | |
| johnston-linear-matrix-algebra_FO2183 | 224 | 1.000 | \operatorname{rank}(A) \leq r | | |
| johnston-linear-matrix-algebra_FO2184 | 224 | 1.000 | A=P \hat{R} | | |
| johnston-linear-matrix-algebra_FO2185 | 224 | 1.000 | \hat{R} | | |
| johnston-linear-matrix-algebra_FO2186 | 225 | 1.000 | C^{T} | | |
| johnston-linear-matrix-algebra_FO2187 | 225 | 1.000 | \hat{R}=E A | | |
| johnston-linear-matrix-algebra_FO2188 | 225 | 1.000 | P=E^{-1} | | |
| johnston-linear-matrix-algebra_FO2189 | 225 | 1.000 | \operatorname{rank}(A)=2 | | |
| johnston-linear-matrix-algebra_FO2190 | 225 | 1.000 | m r | | |
| johnston-linear-matrix-algebra_FO2191 | 225 | 1.000 | r n | | |
| johnston-linear-matrix-algebra_FO2192 | 225 | 0.998 | 11 \times 11=121 | | |
| johnston-linear-matrix-algebra_FO2193 | 225 | 1.000 | (11 \times 2)+(2 \times 11)=44 | | |
| johnston-linear-matrix-algebra_FO2194 | 225 | 1.000 | r=1 | | |
| johnston-linear-matrix-algebra_FO2195 | 226 | 1.000 | \left\{\mathbf{v}_{j}\right\}_{j=1}^{r} \subset \mathbb{R}^{m} | | |
| johnston-linear-matrix-algebra_FO2196 | 226 | 1.000 | \left\{\mathbf{w}_{j}\right\}_{j=1}^{r} \subset \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2197 | 226 | 1.000 | \left\{\mathbf{v}_{i}\right\}_{i=1}^{r} | | |
| johnston-linear-matrix-algebra_FO2198 | 226 | 1.000 | \left\{\mathbf{w}_{i}\right\}_{i=1}^{r} | | |
| johnston-linear-matrix-algebra_FO2199 | 226 | 1.000 | R \in | | |
| johnston-linear-matrix-algebra_FO2200 | 226 | 1.000 | \mathcal{M}_{r, n} | | |
| johnston-linear-matrix-algebra_FO2201 | 226 | 1.000 | \mathbf{v}_{j} \mathbf{w}_{j}^{T} | | |
| johnston-linear-matrix-algebra_FO2202 | 226 | 1.000 | \left\{\mathbf{v}_{j}\right\}_{j=1}^{r} | | |
| johnston-linear-matrix-algebra_FO2203 | 226 | 1.000 | \left\{\mathbf{w}_{j}\right\}_{j=1}^{r} | | |
| johnston-linear-matrix-algebra_FO2204 | 228 | 0.998 | \operatorname{rank}(B) \leq \operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO2205 | 228 | 1.000 | C \in \mathcal{M}_{m, \ell} | | |
| johnston-linear-matrix-algebra_FO2206 | 228 | 1.000 | \operatorname{rank}(C) \leq \operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO2207 | 228 | 1.000 | \operatorname{range}(C) | | |
| johnston-linear-matrix-algebra_FO2208 | 228 | 0.972 | \operatorname{rank}(C) | | |
| johnston-linear-matrix-algebra_FO2209 | 229 | 1.000 | \operatorname{rank}(C)=\operatorname{rank}\left(C^{T}\right) | | |
| johnston-linear-matrix-algebra_FO2210 | 229 | 1.000 | \operatorname{rank}(B)=3 | | |
| johnston-linear-matrix-algebra_FO2211 | 229 | 0.998 | \mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{m}: | | |
| johnston-linear-matrix-algebra_FO2212 | 229 | 1.000 | B \in \mathcal{M}_{k, \ell} | | |
| johnston-linear-matrix-algebra_FO2213 | 229 | 1.000 | \operatorname{rank}(B) \leq | | |
| johnston-linear-matrix-algebra_FO2214 | 229 | 0.991 | \operatorname{range}\left(B^{T}\right) | | |
| johnston-linear-matrix-algebra_FO2215 | 229 | 1.000 | A=\left[\begin{array}{lllll}1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2216 | 229 | 1.000 | B=\left[\begin{array}{ccccc}1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 & 16 \\ 1 & 2 & 5 & 14 & 41 \\ 1 & 3 & 9 & 27 & 81\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2217 | 229 | 0.937 | \operatorname{rank}(A) \geq 3 | | |
| johnston-linear-matrix-algebra_FO2218 | 229 | 0.937 | 3 \times 5 | | |
| johnston-linear-matrix-algebra_FO2219 | 229 | 0.992 | \operatorname{rank}(B) \geq 3 | | |
| johnston-linear-matrix-algebra_FO2220 | 229 | 1.000 | r \times r | | |
| johnston-linear-matrix-algebra_FO2221 | 229 | 0.999 | r \leq \operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO2222 | 230 | 1.000 | \operatorname{rank}(C)=r | | |
| johnston-linear-matrix-algebra_FO2223 | 230 | 0.999 | \operatorname{rank}\left(C^{T}\right)=r | | |
| johnston-linear-matrix-algebra_FO2224 | 230 | 0.999 | \operatorname{range}\left(C^{T}\right) | | |
| johnston-linear-matrix-algebra_FO2225 | 230 | 1.000 | B \in \mathcal{M}_{r} | | |
| johnston-linear-matrix-algebra_FO2226 | 230 | 1.000 | \operatorname{rank}(B)=r | | |
| johnston-linear-matrix-algebra_FO2227 | 230 | 1.000 | \left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2228 | 230 | 1.000 | \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2229 | 230 | 1.000 | \left[\begin{array}{llll}1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2230 | 230 | 0.644 | \operatorname{rank}(A) \leq 3 | | |
| johnston-linear-matrix-algebra_FO2231 | 230 | 0.999 | \operatorname{rank}(A) \geq 4 | | |
| johnston-linear-matrix-algebra_FO2232 | 230 | 0.994 | (n-1) \times(n-1) | | |
| johnston-linear-matrix-algebra_FO2233 | 230 | 0.998 | \operatorname{rank}(A) \leq 1 | | |
| johnston-linear-matrix-algebra_FO2234 | 230 | 0.999 | P \in \mathcal{M}_{r} | | |
| johnston-linear-matrix-algebra_FO2235 | 230 | 1.000 | \tilde{C}=C P^{-1} | | |
| johnston-linear-matrix-algebra_FO2236 | 230 | 1.000 | \tilde{R}=P R | | |
| johnston-linear-matrix-algebra_FO2237 | 230 | 1.000 | A=\tilde{C} \tilde{R} | | |
| johnston-linear-matrix-algebra_FO2238 | 230 | 0.994 | \operatorname{rank}(A) \leq z | | |
| johnston-linear-matrix-algebra_FO2239 | 230 | 1.000 | A=P D Q | | |
| johnston-linear-matrix-algebra_FO2240 | 230 | 1.000 | Q \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO2241 | 230 | 1.000 | A=P B Q | | |
| johnston-linear-matrix-algebra_FO2242 | 230 | 1.000 | \operatorname{rank}(A)=\operatorname{rank}(B) | | |
| johnston-linear-matrix-algebra_FO2243 | 230 | 1.000 | a_{0}, a_{1}, \ldots, a_{m} | | |
| johnston-linear-matrix-algebra_FO2244 | 230 | 1.000 | V \in \mathcal{M}_{m+1, n+1} | | |
| johnston-linear-matrix-algebra_FO2245 | 230 | 0.934 | \operatorname{rank}(V)=\min \{m+1, n+1\} | | |
| johnston-linear-matrix-algebra_FO2246 | 231 | 1.000 | U | | |
| johnston-linear-matrix-algebra_FO2247 | 231 | 1.000 | L | | |
| johnston-linear-matrix-algebra_FO2248 | 231 | 1.000 | \mathbf{L U} | | |
| johnston-linear-matrix-algebra_FO2249 | 231 | 1.000 | A=L U | | |
| johnston-linear-matrix-algebra_FO2250 | 231 | 1.000 | L \in \mathcal{M}_{m} | | |
| johnston-linear-matrix-algebra_FO2251 | 231 | 1.000 | U \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO2252 | 231 | 1.000 | U=R | | |
| johnston-linear-matrix-algebra_FO2253 | 231 | 1.000 | L=P | | |
| johnston-linear-matrix-algebra_FO2254 | 232 | 1.000 | L=E^{-1} | | |
| johnston-linear-matrix-algebra_FO2255 | 232 | 1.000 | j<i | | |
| johnston-linear-matrix-algebra_FO2256 | 232 | 0.873 | A=\left[\begin{array}{cccc}2 & 1 & -2 & 1 \\ -4 & -4 & 3 & 0 \\ 2 & -5 & -2 & 8\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2257 | 233 | 1.000 | L D^{-1} | | |
| johnston-linear-matrix-algebra_FO2258 | 233 | 1.000 | D U | | |
| johnston-linear-matrix-algebra_FO2259 | 233 | 1.000 | A=\left(L D^{-1}\right)(D U) | | |
| johnston-linear-matrix-algebra_FO2260 | 234 | 0.976 | R_{2}-\mathbf{2} R_{1}, R_{3}-\mathbf{3} R_{1} | | |
| johnston-linear-matrix-algebra_FO2261 | 234 | 0.871 | R_{3}-\mathbf{4} R_{2} | | |
| johnston-linear-matrix-algebra_FO2262 | 234 | 1.000 | \ell_{i, j} | | |
| johnston-linear-matrix-algebra_FO2263 | 234 | 1.000 | j=1 | | |
| johnston-linear-matrix-algebra_FO2264 | 234 | 0.882 | \mathbf{a}_{1}^{T}=\mathbf{u}_{1}^{T} | | |
| johnston-linear-matrix-algebra_FO2265 | 234 | 1.000 | n(n+1) / 2 | | |
| johnston-linear-matrix-algebra_FO2266 | 234 | 1.000 | n(n+1)=n^{2}+n | | |
| johnston-linear-matrix-algebra_FO2267 | 234 | 1.000 | n^{2} | | |
| johnston-linear-matrix-algebra_FO2268 | 234 | 1.000 | [U \mid E]\left(R_{2}+2 R_{1}, R_{3}-R_{1}\right. | | |
| johnston-linear-matrix-algebra_FO2269 | 234 | 1.000 | R_{3}-3 R_{2} | | |
| johnston-linear-matrix-algebra_FO2270 | 234 | 1.000 | A, U \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO2271 | 234 | 1.000 | \mathbf{a}_{i}^{T} | | |
| johnston-linear-matrix-algebra_FO2272 | 234 | 1.000 | \mathbf{u}_{i}^{T} | | |
| johnston-linear-matrix-algebra_FO2273 | 234 | 1.000 | R_{i}+\ell_{i, j} R_{j} | | |
| johnston-linear-matrix-algebra_FO2274 | 235 | 1.000 | R_{i}-\ell_{i, j} R_{j} | | |
| johnston-linear-matrix-algebra_FO2275 | 235 | 1.000 | R_{2}-2 R_{1}, R_{3}+3 R_{1} | | |
| johnston-linear-matrix-algebra_FO2276 | 235 | 1.000 | R_{4}+R_{1} | | |
| johnston-linear-matrix-algebra_FO2277 | 235 | 1.000 | \mathbf{a}_{m}^{T} | | |
| johnston-linear-matrix-algebra_FO2278 | 235 | 1.000 | R_{m-1} | | |
| johnston-linear-matrix-algebra_FO2279 | 235 | 1.000 | R_{m-2} | | |
| johnston-linear-matrix-algebra_FO2280 | 236 | 1.000 | i>j | | |
| johnston-linear-matrix-algebra_FO2281 | 236 | 1.000 | L=I | | |
| johnston-linear-matrix-algebra_FO2282 | 236 | 1.000 | U=A | | |
| johnston-linear-matrix-algebra_FO2283 | 236 | 1.000 | i<j | | |
| johnston-linear-matrix-algebra_FO2284 | 236 | 1.000 | L U \mathbf{x}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO2285 | 236 | 1.000 | L \mathbf{y}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO2286 | 236 | 1.000 | U \mathbf{x}=\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2287 | 237 | 1.000 | A \in | | |
| johnston-linear-matrix-algebra_FO2288 | 237 | 1.000 | 2 n^{3} / 3 | | |
| johnston-linear-matrix-algebra_FO2289 | 237 | 1.000 | L \mathbf{y}= | | |
| johnston-linear-matrix-algebra_FO2290 | 237 | 1.000 | 2 n^{2} | | |
| johnston-linear-matrix-algebra_FO2291 | 237 | 1.000 | \mathbf{y}=(0,2,-6,-15) | | |
| johnston-linear-matrix-algebra_FO2292 | 238 | 0.951 | \mathbf{x}=(w, x, y, z)=(22,-9,3,5) | | |
| johnston-linear-matrix-algebra_FO2293 | 238 | 0.593 | \{(0,3),(1,-1),(2,-3),(3,9)\} | | |
| johnston-linear-matrix-algebra_FO2294 | 238 | 1.000 | \{(0,-1),(1,-1),(2,-1),(3,5)\} | | |
| johnston-linear-matrix-algebra_FO2295 | 238 | 1.000 | p(x)=c_{3} x^{3}+c_{2} x^{2}+c_{1} x+c_{0} | | |
| johnston-linear-matrix-algebra_FO2296 | 238 | 1.000 | p(0)=3, p(1)=0, p(2)=1 | | |
| johnston-linear-matrix-algebra_FO2297 | 238 | 1.000 | p(3)=18 | | |
| johnston-linear-matrix-algebra_FO2298 | 239 | 1.000 | L \mathbf{y}=\mathbf{b}=(3,-1,3,9) | | |
| johnston-linear-matrix-algebra_FO2299 | 239 | 1.000 | \mathbf{y}=(3,-4,2,12) | | |
| johnston-linear-matrix-algebra_FO2300 | 239 | 1.000 | U \mathbf{x}=\mathbf{y}=(3,-4,2,12) | | |
| johnston-linear-matrix-algebra_FO2301 | 239 | 1.000 | \mathbf{x}=\left(c_{0}, c_{1}, c_{2}, c_{3}\right)=(3,-1,-5,2) | | |
| johnston-linear-matrix-algebra_FO2302 | 239 | 1.000 | x=0,1,2 | | |
| johnston-linear-matrix-algebra_FO2303 | 239 | 1.000 | L \mathbf{y}=\mathbf{b}=(-1,-1,-1,5) | | |
| johnston-linear-matrix-algebra_FO2304 | 239 | 1.000 | \mathbf{y}=(-1,0,0,6) | | |
| johnston-linear-matrix-algebra_FO2305 | 239 | 1.000 | U \mathbf{x}=\mathbf{y}=(-1,0,0,6) | | |
| johnston-linear-matrix-algebra_FO2306 | 239 | 1.000 | \mathbf{x}=\left(c_{0}, c_{1}, c_{2}, c_{3}\right)=(-1,2,-3,1) | | |
| johnston-linear-matrix-algebra_FO2307 | 239 | 0.817 | (0,-1),(1,-1),(2,-1) | | |
| johnston-linear-matrix-algebra_FO2308 | 239 | 0.817 | (3,5) | | |
| johnston-linear-matrix-algebra_FO2309 | 240 | 1.000 | \mathbf{x}=A^{-1} \mathbf{b}, \mathbf{y}=A^{-1} \mathbf{c} | | |
| johnston-linear-matrix-algebra_FO2310 | 240 | 1.000 | \mathbf{z}=A^{-1} \mathbf{d} | | |
| johnston-linear-matrix-algebra_FO2311 | 240 | 1.000 | 2 n^{3} | | |
| johnston-linear-matrix-algebra_FO2312 | 241 | 1.000 | A=E^{-1} R | | |
| johnston-linear-matrix-algebra_FO2313 | 241 | 1.000 | E^{-1} | | |
| johnston-linear-matrix-algebra_FO2314 | 242 | 1.000 | P=I | | |
| johnston-linear-matrix-algebra_FO2315 | 242 | 0.793 | 0) | | |
| johnston-linear-matrix-algebra_FO2316 | 242 | 0.999 | A=E^{-1} U=P L U | | |
| johnston-linear-matrix-algebra_FO2317 | 242 | 1.000 | A=P L U | | |
| johnston-linear-matrix-algebra_FO2318 | 242 | 1.000 | P \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO2319 | 242 | 1.000 | P^{-1}=P^{T} | | |
| johnston-linear-matrix-algebra_FO2320 | 242 | 1.000 | [U \mid E] | | |
| johnston-linear-matrix-algebra_FO2321 | 242 | 1.000 | E^{-1}=P L | | |
| johnston-linear-matrix-algebra_FO2322 | 243 | 1.000 | P_{*} \in \mathcal{M}_{m} | | |
| johnston-linear-matrix-algebra_FO2323 | 243 | 1.000 | L_{*} \in \mathcal{M}_{m} | | |
| johnston-linear-matrix-algebra_FO2324 | 243 | 1.000 | E=L_{*} P_{*} | | |
| johnston-linear-matrix-algebra_FO2325 | 243 | 1.000 | P_{*}^{-1}=P_{*}^{T} | | |
| johnston-linear-matrix-algebra_FO2326 | 243 | 1.000 | L_{*}^{-1} | | |
| johnston-linear-matrix-algebra_FO2327 | 243 | 1.000 | P=P_{*}^{-1} | | |
| johnston-linear-matrix-algebra_FO2328 | 243 | 1.000 | L=L_{*}^{-1} | | |
| johnston-linear-matrix-algebra_FO2329 | 243 | 1.000 | E P | | |
| johnston-linear-matrix-algebra_FO2330 | 243 | 1.000 | P^{-1} E^{-1} | | |
| johnston-linear-matrix-algebra_FO2331 | 243 | 1.000 | L=P^{-1} E^{-1} | | |
| johnston-linear-matrix-algebra_FO2332 | 243 | 0.999 | A=\left[\begin{array}{cccc}1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 \\ 1 & 2 & 1 & 1 \\ 0 & 1 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2333 | 243 | 1.000 | P^{T} | | |
| johnston-linear-matrix-algebra_FO2334 | 244 | 1.000 | \mathbf{c}=P^{T} \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO2335 | 244 | 1.000 | P \mathbf{c}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO2336 | 244 | 1.000 | P L U \mathbf{x}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO2337 | 244 | 1.000 | L \mathbf{y}=\mathbf{c} | | |
| johnston-linear-matrix-algebra_FO2338 | 245 | 1.000 | \mathbf{b}=(0,1,2,3) | | |
| johnston-linear-matrix-algebra_FO2339 | 245 | 1.000 | P, L | | |
| johnston-linear-matrix-algebra_FO2340 | 245 | 1.000 | \mathbf{y}=(0,2,1,3 / 2) | | |
| johnston-linear-matrix-algebra_FO2341 | 245 | 1.000 | \mathbf{x}=(-3,2,0,1) | | |
| johnston-linear-matrix-algebra_FO2342 | 245 | 0.840 | 1 \leq k \leq r | | |
| johnston-linear-matrix-algebra_FO2343 | 245 | 0.835 | a_{1,1} | | |
| johnston-linear-matrix-algebra_FO2344 | 245 | 1.000 | L=[1] | | |
| johnston-linear-matrix-algebra_FO2345 | 245 | 1.000 | U=\left[a_{1,1}\right] | | |
| johnston-linear-matrix-algebra_FO2346 | 246 | 1.000 | L U=\left(L D^{-1}\right)(D U) | | |
| johnston-linear-matrix-algebra_FO2347 | 246 | 0.743 | B \in \mathcal{M}_{n-1} | | |
| johnston-linear-matrix-algebra_FO2348 | 246 | 1.000 | B=L U | | |
| johnston-linear-matrix-algebra_FO2349 | 246 | 1.000 | \mathbf{x}, \mathbf{y} \in \mathbb{R}^{n-1} | | |
| johnston-linear-matrix-algebra_FO2350 | 246 | 1.000 | d \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2351 | 246 | 1.000 | c=d+\mathbf{x}^{T} \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2352 | 246 | 1.000 | d=c-\mathbf{x}^{T} \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO2353 | 246 | 0.989 | L \mathbf{y}=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2354 | 246 | 1.000 | \mathbf{y}=L^{-1} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2355 | 246 | 1.000 | \mathbf{w}^{T}=\mathbf{x}^{T} U | | |
| johnston-linear-matrix-algebra_FO2356 | 246 | 0.999 | U=B L^{-1} | | |
| johnston-linear-matrix-algebra_FO2357 | 246 | 1.000 | \mathbf{x}=\left(U^{-1}\right)^{T} \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO2358 | 246 | 1.000 | \operatorname{rank}(A)= | | |
| johnston-linear-matrix-algebra_FO2359 | 246 | 0.987 | \mathbf{w}^{T} | | |
| johnston-linear-matrix-algebra_FO2360 | 246 | 1.000 | \operatorname{rank}(A)>r | | |
| johnston-linear-matrix-algebra_FO2361 | 246 | 1.000 | \mathbf{x}^{T} | | |
| johnston-linear-matrix-algebra_FO2362 | 246 | 0.997 | d, \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO2363 | 246 | 0.523 | A=\left[\begin{array}{ccc}3 & -2 & -1 \\ 3 & -1 & 4 \\ 1 & 5 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2364 | 247 | 0.852 | \left[\begin{array}{ll}1 & 2 \\ 2 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2365 | 247 | 1.000 | \left[\begin{array}{cc}2 & -1 \\ 4 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2366 | 247 | 0.995 | \left[\begin{array}{ccc}3 & 1 & 2 \\ -3 & -3 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2367 | 247 | 1.000 | \left[\begin{array}{ccc}1 & 2 & -1 \\ -1 & -3 & -2 \\ 3 & 5 & -8\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2368 | 247 | 0.546 | \left[\begin{array}{ccc}1 & -4 & 5 \\ 3 & -9 & 8 \\ -2 & 5 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2369 | 247 | 1.000 | \left[\begin{array}{cccc}2 & -1 & 4 & 3 \\ -4 & 4 & -7 & -6 \\ 6 & -7 & 12 & 10\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2370 | 247 | 0.947 | \left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]\left[\begin{array}{ll}2 & 3 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}1 \\ -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2371 | 247 | 0.518 | \left[\begin{array}{ccc}1 & 0 & 0 \\ 2 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]\left[\begin{array}{lll}2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2\end{array}\right]\left[\begin{array}{c}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}3 \\ 1 \\ 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2372 | 247 | 0.556 | \left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{c}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2373 | 247 | 0.917 | \left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 2 & 1 & 1 & 0 \\ 1 & 1 & 3 & 1\end{array}\right]\left[\begin{array}{llll}2 & 1 & 0 & 1 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 3\end{array}\right]\left[\begin{array}{c}w \\ x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2374 | 247 | 0.661 | \left[\begin{array}{ll}0 & 2 \\ 1 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2375 | 247 | 0.950 | \left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 2 \\ 1 & 2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2376 | 247 | 0.999 | \left[\begin{array}{llll}0 & 2 & 3 & 3 \\ 1 & 2 & 1 & 2 \\ 1 & 2 & 1 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2377 | 248 | 0.584 | \left[\begin{array}{ll}1 & 3 \\ 3 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2378 | 248 | 1.000 | \left[\begin{array}{cc}0 & 1 \\ 1 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2379 | 248 | 0.997 | \left[\begin{array}{llll}4 & 5 & 5 & 0 \\ 3 & 4 & 3 & 3 \\ 1 & 1 & 2 & 3 \\ 3 & 3 & 4 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2380 | 248 | 1.000 | A=L D U | | |
| johnston-linear-matrix-algebra_FO2381 | 248 | 1.000 | A=L D L^{T} | | |
| johnston-linear-matrix-algebra_FO2382 | 248 | 0.994 | A=\left[\begin{array}{ll}0 & 0 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2386 | 249 | 1.000 | \mathbf{v}_{1}=(1,1) | | |
| johnston-linear-matrix-algebra_FO2387 | 249 | 1.000 | \mathbf{v}_{2}=(-1,1) | | |
| johnston-linear-matrix-algebra_FO2388 | 250 | 1.000 | (1,0) | | |
| johnston-linear-matrix-algebra_FO2389 | 250 | 0.993 | \left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\} | | |
| johnston-linear-matrix-algebra_FO2390 | 250 | 1.000 | \mathbb{R}^{2}=\operatorname{span}((1,0),(0,1),(1,1)) | | |
| johnston-linear-matrix-algebra_FO2391 | 250 | 0.985 | (1,0),(0,1) | | |
| johnston-linear-matrix-algebra_FO2392 | 250 | 1.000 | \{(1,0),(0,1),(1,1)\} | | |
| johnston-linear-matrix-algebra_FO2393 | 250 | 1.000 | (1,1) | | |
| johnston-linear-matrix-algebra_FO2394 | 250 | 0.975 | \{(1,0),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO2395 | 251 | 0.998 | (2,1)=2(1,0)+(0,1)) | | |
| johnston-linear-matrix-algebra_FO2396 | 252 | 1.000 | \mathbb{R}^{n}, B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO2397 | 252 | 1.000 | \left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\} | | |
| johnston-linear-matrix-algebra_FO2398 | 252 | 1.000 | \left\{\mathbf{e}_{2}, \mathbf{e}_{1}\right\} | | |
| johnston-linear-matrix-algebra_FO2399 | 252 | 0.992 | \left(c_{1}, c_{2}\right) | | |
| johnston-linear-matrix-algebra_FO2400 | 252 | 0.992 | \left(c_{2}, c_{1}\right) | | |
| johnston-linear-matrix-algebra_FO2401 | 252 | 1.000 | \mathcal{S}=\mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2402 | 252 | 0.999 | c_{1}, c_{2}, \ldots, c_{n} | | |
| johnston-linear-matrix-algebra_FO2403 | 252 | 1.000 | \mathbf{v}=\left(c_{1}, c_{2}, \ldots, c_{n}\right)=[\mathbf{v}]_{B} | | |
| johnston-linear-matrix-algebra_FO2404 | 252 | 1.000 | \mathbf{v}=(5,1), B=\{(2,1),(1,2)\}, \mathcal{S}=\mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO2405 | 252 | 1.000 | \mathbf{v}=(5,4,3), B=\{(1,2,1),(2,1,1)\}, \mathcal{S}=\operatorname{span}(B) | | |
| johnston-linear-matrix-algebra_FO2406 | 253 | 1.000 | (5,1) | | |
| johnston-linear-matrix-algebra_FO2407 | 253 | 1.000 | c_{1}=3, c_{2}=-1 | | |
| johnston-linear-matrix-algebra_FO2408 | 253 | 1.000 | [\mathbf{v}]_{B}=(3,-1) | | |
| johnston-linear-matrix-algebra_FO2409 | 253 | 0.997 | c_{1}=1, c_{2}=2 | | |
| johnston-linear-matrix-algebra_FO2410 | 253 | 0.997 | [\mathbf{v}]_{B}=(1,2) | | |
| johnston-linear-matrix-algebra_FO2411 | 253 | 1.000 | \left[\mathbf{v}_{j}\right]_{B}=\mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO2412 | 253 | 1.000 | 1 \leq j \leq k | | |
| johnston-linear-matrix-algebra_FO2413 | 253 | 1.000 | \mathbf{v}_{j} | | |
| johnston-linear-matrix-algebra_FO2414 | 253 | 1.000 | \mathbf{e}_{1}, \mathbf{v}_{2} | | |
| johnston-linear-matrix-algebra_FO2415 | 254 | 0.844 | \mathcal{S}=\operatorname{span}((1,2,1),(2,1,1)) | | |
| johnston-linear-matrix-algebra_FO2418 | 254 | 1.000 | \mathbb{R}^{85} | | |
| johnston-linear-matrix-algebra_FO2419 | 254 | 1.000 | [\mathbf{v}]_{B} | | |
| johnston-linear-matrix-algebra_FO2420 | 254 | 1.000 | \mathbf{v}=(2,1,-3,1,2) \in \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO2421 | 255 | 1.000 | c_{1}=2 | | |
| johnston-linear-matrix-algebra_FO2422 | 255 | 1.000 | c_{2}=-3 | | |
| johnston-linear-matrix-algebra_FO2423 | 255 | 1.000 | [\mathbf{v}]_{B}=(2,-3) | | |
| johnston-linear-matrix-algebra_FO2424 | 255 | 0.991 | \mathbf{v}=(2,1,-3,1,2) | | |
| johnston-linear-matrix-algebra_FO2425 | 255 | 1.000 | [\mathbf{v}+\mathbf{w}]_{B}=[\mathbf{v}]_{B}+[\mathbf{w}]_{B} | | |
| johnston-linear-matrix-algebra_FO2426 | 255 | 0.997 | [c \mathbf{v}]_{B}=c[\mathbf{v}]_{B} | | |
| johnston-linear-matrix-algebra_FO2427 | 255 | 1.000 | B=\{(1,2,0,2,1),(0,1,1,1,0)\} | | |
| johnston-linear-matrix-algebra_FO2428 | 255 | 1.000 | \mathbf{w}=(1,0,-2,0,1) | | |
| johnston-linear-matrix-algebra_FO2429 | 255 | 1.000 | [2 \mathbf{v}-5 \mathbf{w}]_{B} | | |
| johnston-linear-matrix-algebra_FO2430 | 255 | 1.000 | 2[\mathbf{v}]_{B}-5[\mathbf{w}]_{B} | | |
| johnston-linear-matrix-algebra_FO2431 | 255 | 1.000 | 2 \mathbf{v}-5 \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO2432 | 256 | 1.000 | \mathbf{v}_{1}=(1,0.002) | | |
| johnston-linear-matrix-algebra_FO2433 | 256 | 1.000 | \mathbf{v}_{2}=(1,0.001) | | |
| johnston-linear-matrix-algebra_FO2434 | 256 | 1.000 | [(-1,1)]_{B}= | | |
| johnston-linear-matrix-algebra_FO2435 | 256 | 1.000 | [2 \mathbf{v}-5 \mathbf{w}]_{B}=(-1,4) | | |
| johnston-linear-matrix-algebra_FO2436 | 256 | 1.000 | [\mathbf{w}]_{B} | | |
| johnston-linear-matrix-algebra_FO2437 | 256 | 1.000 | [\mathbf{w}]_{B}=(1,-2) | | |
| johnston-linear-matrix-algebra_FO2438 | 256 | 1.000 | \mathbf{v}_{1}=(1,0.2) | | |
| johnston-linear-matrix-algebra_FO2439 | 256 | 1.000 | \mathbf{v}_{2}=(1,0.1) | | |
| johnston-linear-matrix-algebra_FO2440 | 257 | 1.000 | \mathbf{v} \neq \mathbf{w} \in B | | |
| johnston-linear-matrix-algebra_FO2441 | 257 | 1.000 | \mathbf{v} \in B | | |
| johnston-linear-matrix-algebra_FO2442 | 257 | 0.998 | \|\mathbf{v}\|=\left\|[\mathbf{v}]_{B}\right\| | | |
| johnston-linear-matrix-algebra_FO2443 | 257 | 1.000 | B=\{(1,1),(1,-1)\} | | |
| johnston-linear-matrix-algebra_FO2444 | 257 | 1.000 | C=\{(3,1),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO2445 | 257 | 1.000 | [\mathbf{v}]_{B}=(5,1) | | |
| johnston-linear-matrix-algebra_FO2446 | 257 | 1.000 | [\mathbf{v}]_{C} | | |
| johnston-linear-matrix-algebra_FO2447 | 257 | 1.000 | \mathbf{v}=5(1,1)+(1,-1)=(6,4) | | |
| johnston-linear-matrix-algebra_FO2448 | 257 | 1.000 | \mathbf{v}=(6,4) | | |
| johnston-linear-matrix-algebra_FO2449 | 257 | 1.000 | (6,4)=c_{1}(3,1)+c_{2}(0,1) | | |
| johnston-linear-matrix-algebra_FO2450 | 257 | 1.000 | [\mathbf{v}]_{C}=(2,2) | | |
| johnston-linear-matrix-algebra_FO2451 | 258 | 1.000 | P_{C \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO2452 | 258 | 1.000 | P_{B \rightarrow C} | | |
| johnston-linear-matrix-algebra_FO2453 | 258 | 1.000 | \left[\mathbf{v}_{1}\right]_{C},\left[\mathbf{v}_{2}\right]_{C}, \ldots,\left[\mathbf{v}_{k}\right]_{C} | | |
| johnston-linear-matrix-algebra_FO2454 | 259 | 1.000 | \mathcal{S}=\operatorname{span}(C) | | |
| johnston-linear-matrix-algebra_FO2455 | 259 | 1.000 | P_{C \leftarrow B}[\mathbf{v}]_{B}=[\mathbf{v}]_{C} | | |
| johnston-linear-matrix-algebra_FO2456 | 259 | 1.000 | \left(P_{C \leftarrow B}\right)^{-1}=P_{B \leftarrow C} | | |
| johnston-linear-matrix-algebra_FO2457 | 259 | 1.000 | \mathbf{v}=c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+\cdots+c_{k} \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO2458 | 259 | 1.000 | [\mathbf{v}]_{B}=\left(c_{1}, c_{2}, \ldots, c_{k}\right) | | |
| johnston-linear-matrix-algebra_FO2459 | 259 | 1.000 | P \in \mathcal{M}_{k} | | |
| johnston-linear-matrix-algebra_FO2460 | 259 | 1.000 | P[\mathbf{v}]_{B}=[\mathbf{v}]_{C} | | |
| johnston-linear-matrix-algebra_FO2461 | 259 | 0.999 | \mathbf{v}=\mathbf{v}_{j} | | |
| johnston-linear-matrix-algebra_FO2462 | 259 | 0.999 | [\mathbf{v}]_{B}=\left[\mathbf{v}_{j}\right]_{B}=\mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO2463 | 259 | 1.000 | P[\mathbf{v}]_{B}=P \mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO2464 | 259 | 1.000 | P[\mathbf{v}]_{B}=[\mathbf{v}]_{C}=\left[\mathbf{v}_{j}\right]_{C} | | |
| johnston-linear-matrix-algebra_FO2465 | 259 | 1.000 | \left[\mathbf{v}_{j}\right]_{C} | | |
| johnston-linear-matrix-algebra_FO2466 | 259 | 1.000 | P=P_{C \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO2467 | 259 | 1.000 | P_{B \leftarrow C} P_{C \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO2468 | 259 | 1.000 | P_{B \leftarrow C} P_{C \leftarrow B}=I | | |
| johnston-linear-matrix-algebra_FO2469 | 259 | 0.999 | [\mathbf{v}]_{B}= | | |
| johnston-linear-matrix-algebra_FO2470 | 259 | 1.000 | [\mathbf{w}]_{B}=(-1,4) | | |
| johnston-linear-matrix-algebra_FO2471 | 259 | 1.000 | B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\} | | |
| johnston-linear-matrix-algebra_FO2472 | 259 | 1.000 | C=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}, \mathbf{w}_{3}\right\} | | |
| johnston-linear-matrix-algebra_FO2473 | 259 | 1.000 | [\mathbf{v}]_{B}=(1,2,3) | | |
| johnston-linear-matrix-algebra_FO2474 | 260 | 1.000 | \mathbf{v}_{1}= | | |
| johnston-linear-matrix-algebra_FO2475 | 260 | 0.604 | c_{1} \mathbf{w}_{1}+c_{2} \mathbf{W}_{2}+c_{3} \mathbf{w}_{3} | | |
| johnston-linear-matrix-algebra_FO2476 | 260 | 1.000 | [\mathbf{w}]_{C} | | |
| johnston-linear-matrix-algebra_FO2477 | 260 | 1.000 | [\mathbf{w}]_{B}=(2,0,-1) | | |
| johnston-linear-matrix-algebra_FO2478 | 260 | 1.000 | \left[\mathbf{v}_{1}\right]_{C},\left[\mathbf{v}_{2}\right]_{C} | | |
| johnston-linear-matrix-algebra_FO2479 | 260 | 1.000 | \left[\mathbf{v}_{3}\right]_{C} | | |
| johnston-linear-matrix-algebra_FO2480 | 260 | 1.000 | \mathbf{w}_{1}, \mathbf{w}_{2} | | |
| johnston-linear-matrix-algebra_FO2481 | 260 | 1.000 | \mathbf{w}_{3} | | |
| johnston-linear-matrix-algebra_FO2482 | 260 | 1.000 | c_{1}=1, c_{2}=-1, c_{3}=1 | | |
| johnston-linear-matrix-algebra_FO2483 | 260 | 1.000 | \left[\mathbf{v}_{1}\right]_{C}=(1,-1,1) | | |
| johnston-linear-matrix-algebra_FO2484 | 260 | 1.000 | \left[\mathbf{v}_{2}\right]_{C}=(0,-1,1) | | |
| johnston-linear-matrix-algebra_FO2485 | 260 | 1.000 | \left[\mathbf{v}_{3}\right]_{C}=(0,-2,1) | | |
| johnston-linear-matrix-algebra_FO2486 | 260 | 1.000 | \mathbb{R}^{k} | | |
| johnston-linear-matrix-algebra_FO2487 | 260 | 1.000 | P_{B \leftarrow C} | | |
| johnston-linear-matrix-algebra_FO2488 | 261 | 1.000 | T: \mathbb{R}^{k} \rightarrow \mathbb{R}^{k} | | |
| johnston-linear-matrix-algebra_FO2489 | 261 | 1.000 | \left[\mathbf{v}_{j}\right]_{B} | | |
| johnston-linear-matrix-algebra_FO2490 | 261 | 1.000 | E=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right\} | | |
| johnston-linear-matrix-algebra_FO2491 | 261 | 1.000 | P_{E \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO2492 | 261 | 1.000 | P_{B \leftarrow E} | | |
| johnston-linear-matrix-algebra_FO2493 | 261 | 0.998 | [\mathbf{v}]_{E}=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2494 | 262 | 0.950 | m \neq n) | | |
| johnston-linear-matrix-algebra_FO2495 | 262 | 1.000 | \left[\mathbf{e}_{1}\right]_{B},\left[\mathbf{e}_{2}\right]_{B} | | |
| johnston-linear-matrix-algebra_FO2496 | 262 | 1.000 | \left[\mathbf{e}_{3}\right]_{B} | | |
| johnston-linear-matrix-algebra_FO2497 | 262 | 1.000 | P_{B \leftarrow E}=P_{E \leftarrow B}^{-1} | | |
| johnston-linear-matrix-algebra_FO2498 | 262 | 1.000 | B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\} | | |
| johnston-linear-matrix-algebra_FO2499 | 262 | 1.000 | [T]_{B} \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO2500 | 262 | 1.000 | [T]_{B} | | |
| johnston-linear-matrix-algebra_FO2502 | 262 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} | | |
| johnston-linear-matrix-algebra_FO2506 | 263 | 0.995 | T(x, y)=(2 x+y, x+2 y) | | |
| johnston-linear-matrix-algebra_FO2507 | 263 | 1.000 | B=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\} | | |
| johnston-linear-matrix-algebra_FO2508 | 263 | 1.000 | T(1,1) | | |
| johnston-linear-matrix-algebra_FO2509 | 263 | 1.000 | T(1,-1) | | |
| johnston-linear-matrix-algebra_FO2510 | 265 | 1.000 | P_{C \leftarrow B}[T]_{B} P_{B \leftarrow C} | | |
| johnston-linear-matrix-algebra_FO2511 | 265 | 1.000 | [T]_{C} | | |
| johnston-linear-matrix-algebra_FO2512 | 265 | 1.000 | [T]_{C}[\mathbf{v}]_{C}=[T(\mathbf{v})]_{C} | | |
| johnston-linear-matrix-algebra_FO2513 | 265 | 1.000 | P_{C \leftarrow B}[T]_{B} P_{B \leftarrow C}=[T]_{C} | | |
| johnston-linear-matrix-algebra_FO2514 | 265 | 1.000 | [T(\mathbf{v})]_{C} | | |
| johnston-linear-matrix-algebra_FO2515 | 265 | 1.000 | B=\{(1,2,3),(2,-1,0),(-1,1,2)\} | | |
| johnston-linear-matrix-algebra_FO2516 | 265 | 1.000 | T: \mathbb{R}^{3} \rightarrow | | |
| johnston-linear-matrix-algebra_FO2517 | 266 | 1.000 | [T]_{E} | | |
| johnston-linear-matrix-algebra_FO2518 | 266 | 1.000 | [T]_{B}=P_{B \leftarrow E}[T] P_{E \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO2519 | 266 | 1.000 | A=[T]_{C} | | |
| johnston-linear-matrix-algebra_FO2520 | 266 | 1.000 | B=[T]_{D} | | |
| johnston-linear-matrix-algebra_FO2521 | 266 | 1.000 | B=P_{D \leftarrow C} A P_{C \leftarrow D} | | |
| johnston-linear-matrix-algebra_FO2522 | 266 | 1.000 | P_{C \leftarrow D}=P_{D \leftarrow C}^{-1} | | |
| johnston-linear-matrix-algebra_FO2523 | 267 | 1.000 | A=P B P^{-1} | | |
| johnston-linear-matrix-algebra_FO2524 | 267 | 1.000 | B=P^{-1} A P | | |
| johnston-linear-matrix-algebra_FO2525 | 267 | 1.000 | B=P A P^{-1} | | |
| johnston-linear-matrix-algebra_FO2526 | 267 | 1.000 | P=\left[\begin{array}{cc}1 & 1 \\ 1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2527 | 267 | 1.000 | A=\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2528 | 267 | 1.000 | B=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2529 | 268 | 1.000 | \operatorname{tr}(A) | | |
| johnston-linear-matrix-algebra_FO2530 | 268 | 1.000 | \left[\begin{array}{ccc}3 & 6 & -3 \\ 0 & -2 & 3 \\ 2 & 4 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2531 | 268 | 0.884 | \operatorname{tr}\left(\left[\begin{array}{ll}2 & 2 \\ 4 & 5\end{array}\right]\right)=2+5=7 | | |
| johnston-linear-matrix-algebra_FO2532 | 268 | 1.000 | \operatorname{tr}\left(\left[\begin{array}{ccc}3 & 6 & -3 \\ 0 & -2 & 3 \\ 2 & 4 & -1\end{array}\right]\right)=3-2-1=0 | | |
| johnston-linear-matrix-algebra_FO2533 | 268 | 1.000 | \operatorname{tr}(A+B)=\operatorname{tr}(A)+\operatorname{tr}(B) | | |
| johnston-linear-matrix-algebra_FO2534 | 268 | 1.000 | \operatorname{tr}(c A)=c \operatorname{tr}(A) | | |
| johnston-linear-matrix-algebra_FO2535 | 268 | 1.000 | \operatorname{tr}(A B)=\operatorname{tr}(B A) | | |
| johnston-linear-matrix-algebra_FO2536 | 269 | 1.000 | [A B]_{i, i} | | |
| johnston-linear-matrix-algebra_FO2537 | 269 | 1.000 | (i, i) | | |
| johnston-linear-matrix-algebra_FO2538 | 269 | 1.000 | a_{i, j} b_{j, i}=b_{j, i} a_{i, j} | | |
| johnston-linear-matrix-algebra_FO2539 | 269 | 1.000 | B=\left[\begin{array}{cc}1 & -1 \\ -2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2540 | 269 | 1.000 | \operatorname{tr}(A B) | | |
| johnston-linear-matrix-algebra_FO2541 | 269 | 1.000 | \operatorname{tr}(B A) | | |
| johnston-linear-matrix-algebra_FO2542 | 269 | 0.997 | \operatorname{tr}(A)=\operatorname{tr}(B) | | |
| johnston-linear-matrix-algebra_FO2543 | 270 | 1.000 | B=\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2544 | 270 | 0.991 | \operatorname{rank}(A)=\operatorname{rank}(B)=2 | | |
| johnston-linear-matrix-algebra_FO2545 | 270 | 1.000 | \operatorname{tr}(A)=5 | | |
| johnston-linear-matrix-algebra_FO2546 | 270 | 1.000 | \operatorname{tr}(B)=6 | | |
| johnston-linear-matrix-algebra_FO2547 | 270 | 0.995 | \operatorname{tr}(A B C)=\operatorname{tr}(C A B) | | |
| johnston-linear-matrix-algebra_FO2548 | 270 | 0.856 | \operatorname{tr}(A B C)=\operatorname{tr}(A C B) | | |
| johnston-linear-matrix-algebra_FO2549 | 270 | 1.000 | \operatorname{tr}(A B C)=1 | | |
| johnston-linear-matrix-algebra_FO2550 | 270 | 1.000 | \operatorname{tr}(A C B)=0 | | |
| johnston-linear-matrix-algebra_FO2551 | 270 | 1.000 | A, B, C, D \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO2552 | 270 | 1.000 | \operatorname{tr}(A B C D) | | |
| johnston-linear-matrix-algebra_FO2553 | 270 | 0.848 | A, B, C | | |
| johnston-linear-matrix-algebra_FO2554 | 270 | 0.999 | \mathbf{v}=(6,1), B=\{(3,0),(0,2)\}, \mathcal{S}=\mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO2555 | 270 | 0.991 | \mathbf{v}=(4,2), B=\{(1,1),(1,-1)\}, \mathcal{S}=\mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO2556 | 270 | 0.987 | \mathbf{v}=(2,3,1), B=\{(1,0,-1),(1,2,1)\}, \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO2557 | 270 | 1.000 | x-y+z=0 | | |
| johnston-linear-matrix-algebra_FO2558 | 270 | 1.000 | \mathbf{v}=(1,1,1), B=\{(1,2,3),(-1,0,1),(2,2,1)\}, \mathcal{S}= | | |
| johnston-linear-matrix-algebra_FO2559 | 270 | 1.000 | \mathbf{v}=(2,3) | | |
| johnston-linear-matrix-algebra_FO2560 | 270 | 1.000 | \mathbf{w}=(2,-1) | | |
| johnston-linear-matrix-algebra_FO2561 | 270 | 0.999 | [\mathbf{v}]_{B}=(1,0) | | |
| johnston-linear-matrix-algebra_FO2562 | 270 | 0.992 | [\mathbf{w}]_{B}=(0,1) | | |
| johnston-linear-matrix-algebra_FO2563 | 270 | 0.998 | [\mathbf{w}]_{B}=(-1,2) | | |
| johnston-linear-matrix-algebra_FO2564 | 270 | 1.000 | \mathbf{v}=(1,5,6,8,-9) | | |
| johnston-linear-matrix-algebra_FO2565 | 270 | 1.000 | \mathbf{v} \in \operatorname{span}(B) | | |
| johnston-linear-matrix-algebra_FO2566 | 270 | 0.995 | B=\{(1,2),(3,4)\}, C=\{(1,0),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO2567 | 270 | 1.000 | B=\{(1,0),(0,1)\}, C=\{(2,1),(-4,3)\} | | |
| johnston-linear-matrix-algebra_FO2568 | 270 | 0.964 | B=\{(1,2),(3,4)\}, C=\{(2,1),(-4,3)\} | | |
| johnston-linear-matrix-algebra_FO2569 | 270 | 1.000 | B=\{(4,0,0),(0,4,0),(1,4,1)\} | | |
| johnston-linear-matrix-algebra_FO2570 | 270 | 1.000 | C=\{(1,2,3),(1,0,1),(-1,1,2)\} | | |
| johnston-linear-matrix-algebra_FO2571 | 270 | 0.925 | \{(2,3),(1,-1)\} | | |
| johnston-linear-matrix-algebra_FO2572 | 270 | 0.996 | T\left(v_{1}, v_{2}\right)=\left(v_{1}-v_{2}, v_{1}+3 v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO2573 | 270 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(v_{1}+2 v_{2}, v_{1}+2 v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO2574 | 270 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(-v_{1}+2 v_{2}, 3 v_{1}\right) | | |
| johnston-linear-matrix-algebra_FO2575 | 270 | 1.000 | T\left(v_{1}, v_{2}\right)=\left(4 v_{1}+3 v_{2}, 5 v_{1}+v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO2576 | 271 | 1.000 | B=\{(3,1,2),(0,-1,-1),(2,-1,0)\} | | |
| johnston-linear-matrix-algebra_FO2577 | 271 | 0.595 | T\left(v_{1}, v_{2}, v_{3}\right)= | | |
| johnston-linear-matrix-algebra_FO2578 | 271 | 0.802 | \begin{aligned} & A=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right] \text { and } B=\left[\begin{array}{cc}2 & -1 \\ -1 & 0\end{array}\right] \\ & \text { (b) } A=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right] \text { and } B=\left[\begin{array}{cc}1 & -2 \\ -2 & 4\end{array}\right]\end{aligned} | | |
| johnston-linear-matrix-algebra_FO2579 | 271 | 0.816 | A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2580 | 271 | 0.816 | B=\left[\begin{array}{ccc}1 & -1 & 0 \\ 3 & 1 & 2 \\ 1 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2581 | 271 | 1.000 | A=\left[\begin{array}{ccc}2 & 0 & 1 \\ 1 & 1 & 3 \\ -1 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2582 | 271 | 1.000 | B=\left[\begin{array}{ccc}1 & -1 & 1 \\ -1 & 1 & 1 \\ 0 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2583 | 271 | 0.997 | A=\left[\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2584 | 271 | 0.997 | B=\left[\begin{array}{cc}3 & -2 \\ 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2585 | 271 | 1.000 | A=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2586 | 271 | 1.000 | B=\left[\begin{array}{ll}2 & 0 \\ 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2587 | 271 | 0.995 | P_{B \leftarrow B}=I | | |
| johnston-linear-matrix-algebra_FO2588 | 271 | 1.000 | \operatorname{tr}(A B)=\operatorname{tr}(A) \operatorname{tr}(B) | | |
| johnston-linear-matrix-algebra_FO2589 | 271 | 1.000 | \operatorname{tr}\left(A^{T}\right)=\operatorname{tr}(A) | | |
| johnston-linear-matrix-algebra_FO2590 | 271 | 1.000 | C=\left\{\mathbf{w}_{1}, \ldots, \mathbf{w}_{m}\right\} \subset \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO2591 | 271 | 1.000 | D=\left\{\left[\mathbf{w}_{1}\right]_{B}, \ldots,\left[\mathbf{w}_{m}\right]_{B}\right\} | | |
| johnston-linear-matrix-algebra_FO2592 | 271 | 1.000 | P=P_{B \leftarrow C} | | |
| johnston-linear-matrix-algebra_FO2593 | 271 | 0.997 | B, C | | |
| johnston-linear-matrix-algebra_FO2594 | 271 | 1.000 | P_{C \leftarrow B}=P_{C \leftarrow E} P_{E \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO2595 | 271 | 1.000 | \left[P_{E \leftarrow C} \mid P_{E \leftarrow B}\right] | | |
| johnston-linear-matrix-algebra_FO2596 | 271 | 1.000 | \left[I \mid P_{C \leftarrow B}\right] | | |
| johnston-linear-matrix-algebra_FO2597 | 271 | 0.998 | \operatorname{nullity}(A)=\operatorname{nullity}(B) | | |
| johnston-linear-matrix-algebra_FO2598 | 272 | 0.999 | f: \mathcal{M}_{n} \rightarrow \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2599 | 272 | 1.000 | f(A B C)=f(A C B) | | |
| johnston-linear-matrix-algebra_FO2600 | 272 | 1.000 | A, B, C \in | | |
| johnston-linear-matrix-algebra_FO2601 | 272 | 1.000 | f(A)=0 | | |
| johnston-linear-matrix-algebra_FO2602 | 272 | 1.000 | f\left(\mathbf{e}_{i} \mathbf{e}_{j}^{T}\right) | | |
| johnston-linear-matrix-algebra_FO2603 | 272 | 1.000 | \mathbf{v} \cdot \mathbf{w}=[\mathbf{v}]_{B} \cdot[\mathbf{w}]_{B} | | |
| johnston-linear-matrix-algebra_FO2604 | 272 | 1.000 | B \subset \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2605 | 272 | 1.000 | \left\{R^{\theta}\left(\mathbf{e}_{1}\right), R^{\theta}\left(\mathbf{e}_{2}\right)\right\} | | |
| johnston-linear-matrix-algebra_FO2606 | 272 | 1.000 | A \mathbf{x} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2607 | 272 | 1.000 | A: A \mathbf{e}_{1}, A \mathbf{e}_{2}, \ldots, A \mathbf{e}_{n} | | |
| johnston-linear-matrix-algebra_FO2608 | 272 | 1.000 | \operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO2609 | 273 | 1.000 | \operatorname{det}(A)=2 | | |
| johnston-linear-matrix-algebra_FO2611 | 273 | 1.000 | \operatorname{det}(I)=1 | | |
| johnston-linear-matrix-algebra_FO2612 | 273 | 1.000 | \operatorname{det}(B) | | |
| johnston-linear-matrix-algebra_FO2613 | 273 | 1.000 | n=2 | | |
| johnston-linear-matrix-algebra_FO2614 | 274 | 1.000 | \mathbf{a}_{1} | | |
| johnston-linear-matrix-algebra_FO2618 | 274 | 0.998 | \operatorname{det}: \mathcal{M}_{n} \rightarrow \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2619 | 274 | 1.000 | \operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B) | | |
| johnston-linear-matrix-algebra_FO2620 | 274 | 1.000 | \mathbf{v}, \mathbf{w}, \mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO2621 | 275 | 1.000 | \operatorname{det}(A) \neq 0 | | |
| johnston-linear-matrix-algebra_FO2622 | 275 | 1.000 | \operatorname{det}\left(A^{-1}\right)=1 / \operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO2623 | 275 | 1.000 | \operatorname{det}(A)=0 | | |
| johnston-linear-matrix-algebra_FO2624 | 275 | 1.000 | \operatorname{det}\left(P^{-1} A P\right)=0 | | |
| johnston-linear-matrix-algebra_FO2625 | 275 | 0.817 | P \mathbf{e}_{1}=\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO2626 | 275 | 1.000 | P^{-1} A P | | |
| johnston-linear-matrix-algebra_FO2627 | 276 | 1.000 | \operatorname{det}\left(P^{-1} A P\right)=\operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO2628 | 276 | 0.965 | \operatorname{det}(A)=\operatorname{det}(B) | | |
| johnston-linear-matrix-algebra_FO2629 | 276 | 1.000 | \operatorname{det}(c A)=c^{n} \operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO2630 | 276 | 1.000 | \operatorname{det}\left(A^{T}\right)=\operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO2631 | 276 | 1.000 | c^{n} | | |
| johnston-linear-matrix-algebra_FO2633 | 277 | 0.998 | A, B, C \in \mathcal{M}_{3} | | |
| johnston-linear-matrix-algebra_FO2634 | 277 | 0.998 | \operatorname{det}(A)=2, \operatorname{det}(B)=3 | | |
| johnston-linear-matrix-algebra_FO2635 | 277 | 1.000 | \operatorname{det}(C)=5 | | |
| johnston-linear-matrix-algebra_FO2636 | 277 | 1.000 | \operatorname{det}(A B) | | |
| johnston-linear-matrix-algebra_FO2637 | 277 | 1.000 | \operatorname{det}\left(A^{2} C^{T} B^{-1}\right) | | |
| johnston-linear-matrix-algebra_FO2638 | 277 | 1.000 | \operatorname{det}\left(3 A B^{-2} C^{2}\right) | | |
| johnston-linear-matrix-algebra_FO2639 | 277 | 1.000 | \operatorname{det}\left(2 A^{-3} B^{-2}\left(C^{T} B\right)^{4}(C / 5)^{2}\right) | | |
| johnston-linear-matrix-algebra_FO2640 | 277 | 1.000 | \operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)=2 \cdot 3=6 | | |
| johnston-linear-matrix-algebra_FO2641 | 277 | 1.000 | \operatorname{det}\left(A^{2} C^{T} B^{-1}\right)=\operatorname{det}(A)^{2} \operatorname{det}\left(C^{T}\right) \operatorname{det}\left(B^{-1}\right)=2^{2} \cdot 5 \cdot(1 / 3)=20 / 3 | | |
| johnston-linear-matrix-algebra_FO2642 | 277 | 1.000 | \operatorname{det}\left(3 A B^{-2} C^{2}\right)=3^{3} \operatorname{det}(A) \operatorname{det}(B)^{-2} \operatorname{det}(C)^{2}=27 \cdot 2 \cdot(1 / 9) \cdot 25= | | |
| johnston-linear-matrix-algebra_FO2643 | 278 | 1.000 | c \mathbf{e}_{i} | | |
| johnston-linear-matrix-algebra_FO2644 | 278 | 0.523 | 1+c \cdot 0=1 | | |
| johnston-linear-matrix-algebra_FO2646 | 279 | 1.000 | \mathbf{e}_{i}+\mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO2647 | 280 | 1.000 | f(x) | | |
| johnston-linear-matrix-algebra_FO2648 | 280 | 1.000 | x=b | | |
| johnston-linear-matrix-algebra_FO2649 | 280 | 1.000 | f(x) \geq 0 | | |
| johnston-linear-matrix-algebra_FO2650 | 281 | 1.000 | c_{1} R_{i_{1}}, c_{2} R_{i_{2}}, \ldots, c_{k} R_{i_{k}} | | |
| johnston-linear-matrix-algebra_FO2651 | 281 | 1.000 | E_{m} \cdots E_{2} E_{1} A=I | | |
| johnston-linear-matrix-algebra_FO2652 | 281 | 0.999 | E_{1}, E_{2}, \ldots, E_{m} | | |
| johnston-linear-matrix-algebra_FO2653 | 282 | 1.000 | 1 /(-1)^{s}=(-1)^{s} | | |
| johnston-linear-matrix-algebra_FO2654 | 282 | 1.000 | \left[\begin{array}{ll}0 & 2 \\ 3 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2655 | 282 | 1.000 | \frac{1}{2} R_{1} | | |
| johnston-linear-matrix-algebra_FO2656 | 282 | 1.000 | 1 /(1 / 2)=2 | | |
| johnston-linear-matrix-algebra_FO2657 | 282 | 1.000 | s=1 | | |
| johnston-linear-matrix-algebra_FO2658 | 282 | 0.999 | \frac{1}{3} R_{1} | | |
| johnston-linear-matrix-algebra_FO2659 | 282 | 0.997 | (-1) /((1 / 2)(1 / 3))=-6 | | |
| johnston-linear-matrix-algebra_FO2660 | 283 | 1.000 | \frac{1}{2} R_{3} | | |
| johnston-linear-matrix-algebra_FO2661 | 283 | 1.000 | a_{1,1}, a_{2,2}, \ldots, a_{n, n} \neq 0 | | |
| johnston-linear-matrix-algebra_FO2662 | 283 | 1.000 | a_{n, n} | | |
| johnston-linear-matrix-algebra_FO2663 | 283 | 1.000 | 1 \leq i \leq n-1 | | |
| johnston-linear-matrix-algebra_FO2664 | 283 | 1.000 | a_{i, n} R_{n} | | |
| johnston-linear-matrix-algebra_FO2665 | 283 | 1.000 | R_{i} | | |
| johnston-linear-matrix-algebra_FO2666 | 283 | 1.000 | a_{n-1, n-1} | | |
| johnston-linear-matrix-algebra_FO2667 | 283 | 1.000 | 1 \leq i \leq n-2 | | |
| johnston-linear-matrix-algebra_FO2668 | 283 | 1.000 | a_{i, n-1} R_{n-1} | | |
| johnston-linear-matrix-algebra_FO2669 | 284 | 0.894 | a_{1,1} a_{2,2} \cdots a_{n, n}= | | |
| johnston-linear-matrix-algebra_FO2670 | 284 | 1.000 | a_{i, i}=0 | | |
| johnston-linear-matrix-algebra_FO2671 | 284 | 0.958 | r_{1,1}, r_{2,2}, \ldots, r_{n, n} | | |
| johnston-linear-matrix-algebra_FO2672 | 284 | 1.000 | \left[\begin{array}{cccc}0 & 2 & 1 & 2 \\ 1 & -1 & 1 & 0 \\ 2 & 1 & 0 & 1 \\ -2 & 0 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2673 | 284 | 1.000 | 2 \cdot 1=2 | | |
| johnston-linear-matrix-algebra_FO2674 | 285 | 1.000 | (1 / 3) R_{1} | | |
| johnston-linear-matrix-algebra_FO2675 | 285 | 0.999 | (1 \cdot(-2) \cdot 1) /(1 / 3)=-6 | | |
| johnston-linear-matrix-algebra_FO2676 | 285 | 1.000 | (-1) \cdot 1 \cdot 2 \cdot(-7 / 2) \cdot(5 / 7)=5 | | |
| johnston-linear-matrix-algebra_FO2677 | 285 | 0.435 | [a] | | |
| johnston-linear-matrix-algebra_FO2678 | 286 | 1.000 | B=\left[\begin{array}{ll}0 & 2 \\ 3 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2679 | 286 | 0.966 | \operatorname{tr}(A)=\operatorname{tr}(B)=5 | | |
| johnston-linear-matrix-algebra_FO2680 | 286 | 1.000 | \operatorname{det}(A) \neq \operatorname{det}(B) | | |
| johnston-linear-matrix-algebra_FO2681 | 288 | 1.000 | a e i+b f g+c d h-a f h-b d i-c e g | | |
| johnston-linear-matrix-algebra_FO2682 | 288 | 1.000 | \left[\begin{array}{ccc}0 & 2 & 1 \\ 1 & -1 & 1 \\ 2 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2683 | 289 | 1.000 | m_{i, j} | | |
| johnston-linear-matrix-algebra_FO2684 | 289 | 1.000 | c_{i, j} | | |
| johnston-linear-matrix-algebra_FO2685 | 289 | 0.898 | (\mathbf{i}, \mathbf{j}) | | |
| johnston-linear-matrix-algebra_FO2686 | 289 | 0.843 | \mathbf{i}, \mathbf{j} | | |
| johnston-linear-matrix-algebra_FO2687 | 289 | 0.843 | (-1)^{i+j} m_{i, j} | | |
| johnston-linear-matrix-algebra_FO2688 | 289 | 0.860 | (-1)^{i+j} | | |
| johnston-linear-matrix-algebra_FO2689 | 289 | 1.000 | c_{1,2}=-m_{1,2}=-(-6)=6 | | |
| johnston-linear-matrix-algebra_FO2690 | 290 | 0.793 | (3,2) | | |
| johnston-linear-matrix-algebra_FO2691 | 291 | 1.000 | \left\{c_{i, j}\right\}_{i, j=1}^{n} | | |
| johnston-linear-matrix-algebra_FO2692 | 293 | 1.000 | c_{1,1} | | |
| johnston-linear-matrix-algebra_FO2693 | 294 | 1.000 | i-1 | | |
| johnston-linear-matrix-algebra_FO2694 | 294 | 1.000 | i-2 | | |
| johnston-linear-matrix-algebra_FO2695 | 294 | 1.000 | i-3 | | |
| johnston-linear-matrix-algebra_FO2696 | 294 | 0.992 | (-1)^{i-1} | | |
| johnston-linear-matrix-algebra_FO2697 | 294 | 0.992 | c_{i, 1} | | |
| johnston-linear-matrix-algebra_FO2698 | 294 | 1.000 | \operatorname{det}(A)=\operatorname{det}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO2699 | 296 | 1.000 | f(I)=2 | | |
| johnston-linear-matrix-algebra_FO2700 | 296 | 1.000 | f(I)=1 | | |
| johnston-linear-matrix-algebra_FO2701 | 296 | 1.000 | f(A B)=f(A) f(B) | | |
| johnston-linear-matrix-algebra_FO2702 | 296 | 1.000 | A=B=I | | |
| johnston-linear-matrix-algebra_FO2703 | 296 | 0.998 | \left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2704 | 296 | 1.000 | \left[\begin{array}{ccc}1 & 3 & 0 \\ 0 & -2 & 2 \\ -1 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2705 | 296 | 0.832 | \left[\begin{array}{ccc}2 & 0 & 1 \\ 5 & 2 & 8 \\ 3 & -2 & 7\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2706 | 296 | 1.000 | \left[\begin{array}{ccc}3 & 2 & -6 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2707 | 296 | 1.000 | \left[\begin{array}{cccc}1 & 2 & 1 & 0 \\ 2 & 1 & 2 & -3 \\ 4 & 3 & 1 & 2 \\ 0 & 0 & 2 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2708 | 297 | 0.994 | \left[\begin{array}{cccc}3 & 1 & 6 & 4 \\ -2 & 5 & 3 & 3 \\ 6 & -2 & 4 & 1 \\ 3 & 4 & -2 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2709 | 297 | 1.000 | \left[\begin{array}{ccccc}-2 & 6 & -1 & 6 & -1 \\ 5 & 4 & -1 & 3 & -1 \\ 4 & 3 & 4 & -2 & 1 \\ 6 & 6 & 5 & 4 & 2 \\ -1 & 2 & -2 & -1 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2710 | 297 | 0.999 | \left[\begin{array}{cccccc}2 & 2 & 5 & -1 & 1 & -1 \\ 1 & 3 & 0 & 0 & 1 & 1 \\ 1 & 2 & 5 & -1 & 0 & 3 \\ 4 & 0 & 5 & 5 & -1 & 3 \\ 5 & -1 & 5 & 2 & 0 & -1 \\ 3 & 0 & 3 & 3 & 4 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2711 | 297 | 1.000 | \left[\begin{array}{cccccc}-2 & 3 & 2 & 5 & 2 & 4 \\ -1 & 2 & -1 & -3 & 2 & 0 \\ 6 & -2 & 3 & 5 & 4 & -1 \\ -2 & 2 & -2 & -2 & -3 & 3 \\ 3 & 3 & 2 & 2 & -3 & 1 \\ 1 & 6 & 0 & -1 & -3 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2712 | 297 | 1.000 | A, B, C \in \mathcal{M}_{5} | | |
| johnston-linear-matrix-algebra_FO2713 | 297 | 1.000 | \operatorname{det}(B)=3 | | |
| johnston-linear-matrix-algebra_FO2714 | 297 | 1.000 | \operatorname{det}(C)=0 | | |
| johnston-linear-matrix-algebra_FO2715 | 297 | 0.904 | A B^{T} | | |
| johnston-linear-matrix-algebra_FO2716 | 297 | 1.000 | A^{2} B | | |
| johnston-linear-matrix-algebra_FO2717 | 297 | 1.000 | 2 A^{2} | | |
| johnston-linear-matrix-algebra_FO2718 | 297 | 1.000 | A^{-1} B^{3} | | |
| johnston-linear-matrix-algebra_FO2719 | 297 | 1.000 | A^{3} B^{-2} A^{-4} B^{2} A | | |
| johnston-linear-matrix-algebra_FO2720 | 297 | 1.000 | 3 A^{2}\left(B^{T} B / 2\right)^{-1} | | |
| johnston-linear-matrix-algebra_FO2721 | 297 | 1.000 | A^{64} B^{-14} C^{9} A^{-6} B^{13} | | |
| johnston-linear-matrix-algebra_FO2722 | 297 | 1.000 | \operatorname{det}(A)=6 | | |
| johnston-linear-matrix-algebra_FO2723 | 297 | 1.000 | 3 R_{3} | | |
| johnston-linear-matrix-algebra_FO2724 | 297 | 0.961 | R_{1} \leftrightarrow R_{4} | | |
| johnston-linear-matrix-algebra_FO2725 | 297 | 1.000 | R_{2}-2 R_{1}, R_{3}-4 R_{1} | | |
| johnston-linear-matrix-algebra_FO2726 | 297 | 0.991 | 2 R_{1}, R_{2} \leftrightarrow R_{4} | | |
| johnston-linear-matrix-algebra_FO2727 | 297 | 1.000 | R_{1} \leftrightarrow R_{2}, R_{2} \leftrightarrow R_{3} | | |
| johnston-linear-matrix-algebra_FO2728 | 297 | 1.000 | 2 R_{2}, 3 R_{3}, 4 R_{4} | | |
| johnston-linear-matrix-algebra_FO2729 | 297 | 1.000 | R_{1} \leftrightarrow R_{3}, 2 R_{2}, R_{3}-3 R_{1}, R_{4}-7 R_{2}, 3 R_{4} | | |
| johnston-linear-matrix-algebra_FO2730 | 297 | 0.979 | \left[\begin{array}{lll}g & h & i \\ d & e & f \\ a & b & c\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2731 | 297 | 1.000 | \left[\begin{array}{ccc}a & b & c \\ 2 d & 2 e & 2 f \\ 3 g & 3 h & 3 i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2732 | 297 | 0.566 | \left[\begin{array}{ccc}a & b+a & 2 c \\ d & e+d & 2 f \\ g & h+g & 2 i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2733 | 297 | 1.000 | \left[\begin{array}{ccc}a & 2 b & 3 c \\ 2 d & 4 e & 6 f \\ 3 g & 6 h & 9 i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2734 | 297 | 0.997 | \operatorname{det}(-A)= | | |
| johnston-linear-matrix-algebra_FO2735 | 297 | 1.000 | -\operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO2736 | 297 | 1.000 | \operatorname{det}(A B)=\operatorname{det}(B A) | | |
| johnston-linear-matrix-algebra_FO2737 | 297 | 1.000 | \operatorname{det}(A+B)=\operatorname{det}(A)+\operatorname{det}(B) | | |
| johnston-linear-matrix-algebra_FO2738 | 297 | 0.952 | \operatorname{det}(A)=3 | | |
| johnston-linear-matrix-algebra_FO2739 | 297 | 1.000 | \operatorname{det}(A)=1 | | |
| johnston-linear-matrix-algebra_FO2740 | 297 | 1.000 | B_{n} | | |
| johnston-linear-matrix-algebra_FO2741 | 297 | 1.000 | \operatorname{det}\left(B_{n}\right) | | |
| johnston-linear-matrix-algebra_FO2742 | 297 | 0.977 | \left[P_{\mathbf{u}}\right] | | |
| johnston-linear-matrix-algebra_FO2743 | 297 | 1.000 | \operatorname{det}\left(\left[P_{\mathbf{u}}\right]\right) | | |
| johnston-linear-matrix-algebra_FO2744 | 297 | 1.000 | \operatorname{det}\left(A^{r}\right)=(\operatorname{det}(A))^{r} | | |
| johnston-linear-matrix-algebra_FO2745 | 297 | 1.000 | A^{T}=-A | | |
| johnston-linear-matrix-algebra_FO2746 | 298 | 1.000 | A^{R} | | |
| johnston-linear-matrix-algebra_FO2747 | 298 | 1.000 | \operatorname{det}\left(A^{R}\right) | | |
| johnston-linear-matrix-algebra_FO2748 | 298 | 1.000 | \operatorname{det}\left(A_{n}\right) | | |
| johnston-linear-matrix-algebra_FO2749 | 298 | 1.000 | n=2,3,4,5 | | |
| johnston-linear-matrix-algebra_FO2750 | 298 | 0.971 | (*) | | |
| johnston-linear-matrix-algebra_FO2751 | 298 | 1.000 | \prod_{j=1}^{n} | | |
| johnston-linear-matrix-algebra_FO2752 | 298 | 1.000 | A \in \mathcal{M}_{m}, B \in \mathcal{M}_{m, n}, C \in \mathcal{M}_{n, m} | | |
| johnston-linear-matrix-algebra_FO2753 | 298 | 0.922 | \operatorname{det}\left(I_{n}+\mathbf{v} \mathbf{w}^{T}\right)=1+\mathbf{w}^{T} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2754 | 298 | 0.827 | \operatorname{det}\left(A+\mathbf{v w}^{T}\right)=\left(1+\mathbf{w}^{T} A^{-1} \mathbf{v}\right) \operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO2755 | 298 | 0.971 | A+\mathbf{v w}^{T}=A\left(I_{n}+A^{-1} \mathbf{v w}^{T}\right) | | |
| johnston-linear-matrix-algebra_FO2756 | 298 | 1.000 | (-1)^{n+1} | | |
| johnston-linear-matrix-algebra_FO2757 | 298 | 1.000 | f: \mathcal{M}_{n} \rightarrow | | |
| johnston-linear-matrix-algebra_FO2758 | 298 | 1.000 | f(A+c B)= | | |
| johnston-linear-matrix-algebra_FO2759 | 298 | 1.000 | f(A)+c f(B) | | |
| johnston-linear-matrix-algebra_FO2760 | 299 | 1.000 | \lambda | | |
| johnston-linear-matrix-algebra_FO2761 | 299 | 1.000 | A \mathbf{e}_{j}=a_{j, j} \mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO2762 | 299 | 1.000 | a_{1,1}, a_{2,2}, \ldots, a_{n, n} | | |
| johnston-linear-matrix-algebra_FO2763 | 300 | 1.000 | A \mathbf{0}=\lambda \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2764 | 300 | 1.000 | A \mathbf{v}=3 \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2765 | 300 | 1.000 | \lambda=3 | | |
| johnston-linear-matrix-algebra_FO2766 | 300 | 1.000 | \lambda=1 | | |
| johnston-linear-matrix-algebra_FO2767 | 300 | 1.000 | A \mathbf{v}=1 \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2768 | 300 | 1.000 | v_{1}=-v_{2} | | |
| johnston-linear-matrix-algebra_FO2769 | 300 | 1.000 | \mathbf{v}=\left(-v_{2}, v_{2}\right) | | |
| johnston-linear-matrix-algebra_FO2770 | 300 | 1.000 | v_{2}=1 | | |
| johnston-linear-matrix-algebra_FO2771 | 300 | 1.000 | \mathbf{v}=(-1,1) | | |
| johnston-linear-matrix-algebra_FO2772 | 301 | 1.000 | A \mathbf{v}-\lambda \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2773 | 301 | 1.000 | (A-\lambda) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2774 | 301 | 0.955 | A \mathbf{v}=\lambda \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2775 | 301 | 1.000 | (A-\lambda I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2776 | 301 | 1.000 | A-\lambda I | | |
| johnston-linear-matrix-algebra_FO2777 | 301 | 1.000 | \operatorname{det}(A-\lambda I) | | |
| johnston-linear-matrix-algebra_FO2778 | 301 | 1.000 | \operatorname{det}(A-\lambda I)=0 | | |
| johnston-linear-matrix-algebra_FO2779 | 301 | 0.998 | A=\left[\begin{array}{ll}1 & 2 \\ 5 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2780 | 302 | 1.000 | \lambda=-1 | | |
| johnston-linear-matrix-algebra_FO2781 | 302 | 1.000 | \lambda=6 | | |
| johnston-linear-matrix-algebra_FO2782 | 302 | 1.000 | \lambda^{2}-5 \lambda-6 | | |
| johnston-linear-matrix-algebra_FO2783 | 302 | 1.000 | a \lambda^{2}+b \lambda+c=0 | | |
| johnston-linear-matrix-algebra_FO2784 | 302 | 1.000 | \lambda=(5 \pm 7) / 2 | | |
| johnston-linear-matrix-algebra_FO2785 | 302 | 1.000 | \mathbf{v} \in \operatorname{null}(A-\lambda I) | | |
| johnston-linear-matrix-algebra_FO2786 | 302 | 1.000 | (A+I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2787 | 302 | 1.000 | (A-6 I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2788 | 302 | 0.998 | (A-\lambda I) \mathbf{v}= | | |
| johnston-linear-matrix-algebra_FO2789 | 302 | 1.000 | \left(-v_{2}, v_{2}\right)=v_{2}(-1,1) | | |
| johnston-linear-matrix-algebra_FO2790 | 303 | 0.999 | (2 / 5,1) | | |
| johnston-linear-matrix-algebra_FO2791 | 303 | 1.000 | v_{1}=2 v_{2} / 5 | | |
| johnston-linear-matrix-algebra_FO2792 | 303 | 0.999 | \mathbf{v}=\left(2 v_{2} / 5, v_{2}\right)=v_{2}(2 / 5,1) | | |
| johnston-linear-matrix-algebra_FO2793 | 304 | 1.000 | (2-\lambda) | | |
| johnston-linear-matrix-algebra_FO2794 | 304 | 0.816 | \mathbf{v}_{1}=(-1,1) | | |
| johnston-linear-matrix-algebra_FO2795 | 304 | 0.816 | \mathbf{v}_{2}=(2 / 5,1) | | |
| johnston-linear-matrix-algebra_FO2796 | 304 | 1.000 | \lambda=-2, \lambda=2 | | |
| johnston-linear-matrix-algebra_FO2797 | 304 | 1.000 | \lambda=4 | | |
| johnston-linear-matrix-algebra_FO2798 | 304 | 1.000 | \lambda=-2 | | |
| johnston-linear-matrix-algebra_FO2799 | 304 | 1.000 | (A+2 I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2800 | 304 | 1.000 | v_{3}=0 | | |
| johnston-linear-matrix-algebra_FO2801 | 304 | 0.954 | (-1,1,0) | | |
| johnston-linear-matrix-algebra_FO2802 | 304 | 1.000 | \lambda=2 | | |
| johnston-linear-matrix-algebra_FO2803 | 305 | 0.706 | \mathbf{3} \times \mathbf{3} | | |
| johnston-linear-matrix-algebra_FO2804 | 305 | 1.000 | v_{1}=0 | | |
| johnston-linear-matrix-algebra_FO2805 | 305 | 1.000 | v_{2}=-v_{3} | | |
| johnston-linear-matrix-algebra_FO2806 | 305 | 1.000 | (A-4 I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2807 | 305 | 1.000 | A-4 I | | |
| johnston-linear-matrix-algebra_FO2808 | 305 | 1.000 | v_{1}=v_{2} | | |
| johnston-linear-matrix-algebra_FO2809 | 305 | 1.000 | A=\left[\begin{array}{ccc}1 & 2 & 3 \\ 1 & -2 & 1 \\ 3 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2810 | 305 | 1.000 | -\lambda^{3}+16 \lambda+24=0 | | |
| johnston-linear-matrix-algebra_FO2811 | 306 | 0.996 | -(-2)^{3}+16(-2)+ | | |
| johnston-linear-matrix-algebra_FO2812 | 306 | 1.000 | 24=8-32+24=0 | | |
| johnston-linear-matrix-algebra_FO2813 | 306 | 1.000 | -\lambda^{3}+16 \lambda+ | | |
| johnston-linear-matrix-algebra_FO2814 | 306 | 1.000 | \lambda+2 | | |
| johnston-linear-matrix-algebra_FO2815 | 306 | 0.990 | -\lambda^{3}+16 \lambda+24=(\lambda+2)\left(-\lambda^{2}+2 \lambda+12\right) | | |
| johnston-linear-matrix-algebra_FO2816 | 306 | 1.000 | -\lambda^{3}+16 \lambda+24 | | |
| johnston-linear-matrix-algebra_FO2817 | 306 | 1.000 | -\lambda^{2}+2 \lambda+12 | | |
| johnston-linear-matrix-algebra_FO2818 | 306 | 0.998 | \lambda=-2, \lambda=1+\sqrt{13} | | |
| johnston-linear-matrix-algebra_FO2819 | 306 | 0.998 | \lambda=1- | | |
| johnston-linear-matrix-algebra_FO2820 | 306 | 1.000 | \sqrt{13} | | |
| johnston-linear-matrix-algebra_FO2821 | 306 | 1.000 | A- | | |
| johnston-linear-matrix-algebra_FO2822 | 306 | 1.000 | \lambda I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2823 | 306 | 0.921 | 5 \times 5 | | |
| johnston-linear-matrix-algebra_FO2824 | 307 | 1.000 | \operatorname{det}(\lambda I-A) | | |
| johnston-linear-matrix-algebra_FO2825 | 307 | 1.000 | p_{A}: \mathbb{R} \rightarrow \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2826 | 307 | 1.000 | \left(a_{1,1}-\lambda\right)\left(a_{2,2}-\lambda\right) \cdots\left(a_{n, n}-\lambda\right) | | |
| johnston-linear-matrix-algebra_FO2827 | 307 | 1.000 | p_{A} | | |
| johnston-linear-matrix-algebra_FO2828 | 307 | 1.000 | \lambda_{0} | | |
| johnston-linear-matrix-algebra_FO2829 | 307 | 1.000 | \left(\lambda_{0}-\lambda\right) | | |
| johnston-linear-matrix-algebra_FO2830 | 307 | 1.000 | p_{A}(\lambda) | | |
| johnston-linear-matrix-algebra_FO2831 | 308 | 1.000 | \sqrt{-1}=i | | |
| johnston-linear-matrix-algebra_FO2832 | 309 | 1.000 | a+b i | | |
| johnston-linear-matrix-algebra_FO2833 | 309 | 1.000 | \overline{a+b i}=a-b i | | |
| johnston-linear-matrix-algebra_FO2834 | 309 | 1.000 | \mathcal{M}_{m, n}(X) | | |
| johnston-linear-matrix-algebra_FO2835 | 309 | 1.000 | \mathcal{M}_{n}(X) | | |
| johnston-linear-matrix-algebra_FO2836 | 309 | 0.999 | \overline{-1+2 i}=-1-2 i | | |
| johnston-linear-matrix-algebra_FO2837 | 309 | 1.000 | p_{A}(\lambda)=0 | | |
| johnston-linear-matrix-algebra_FO2838 | 309 | 1.000 | p_{A}(\bar{\lambda})=\overline{p_{A}(\lambda)}=0 | | |
| johnston-linear-matrix-algebra_FO2839 | 309 | 1.000 | \mathcal{M}_{m, n}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO2840 | 309 | 1.000 | \mathcal{M}_{m, n}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO2841 | 309 | 1.000 | p_{A}(\lambda)=p_{B}(\lambda) | | |
| johnston-linear-matrix-algebra_FO2842 | 310 | 1.000 | \lambda^{n} | | |
| johnston-linear-matrix-algebra_FO2843 | 310 | 0.990 | (-1)^{n} | | |
| johnston-linear-matrix-algebra_FO2844 | 310 | 1.000 | A=\left[\begin{array}{ccc}4 & 0 & 2 \\ -2 & 6 & 2 \\ 2 & 0 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2845 | 310 | 1.000 | B=\left[\begin{array}{ccc}8 & -5 & -5 \\ 5 & -2 & -5 \\ -5 & 5 & 8\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2846 | 310 | 0.975 | \operatorname{rank}(A)=\operatorname{rank}(B)=3, \operatorname{tr}(A)=\operatorname{tr}(B)=14 | | |
| johnston-linear-matrix-algebra_FO2847 | 310 | 0.975 | \operatorname{det}(A)=\operatorname{det}(B)=72 | | |
| johnston-linear-matrix-algebra_FO2848 | 310 | 1.000 | p_{A}(\lambda) \neq p_{B}(\lambda) | | |
| johnston-linear-matrix-algebra_FO2849 | 310 | 1.000 | \lambda^{2} | | |
| johnston-linear-matrix-algebra_FO2850 | 310 | 1.000 | A \in \mathcal{M}_{n}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO2851 | 310 | 1.000 | \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} | | |
| johnston-linear-matrix-algebra_FO2852 | 310 | 1.000 | c_{0}=\operatorname{det}(A)=\lambda_{1} \lambda_{2} \cdots \lambda_{n} | | |
| johnston-linear-matrix-algebra_FO2853 | 311 | 1.000 | \lambda=0 | | |
| johnston-linear-matrix-algebra_FO2854 | 311 | 1.000 | (-1)^{n-1} c_{n-1}=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{n} | | |
| johnston-linear-matrix-algebra_FO2855 | 311 | 1.000 | \lambda^{n-1} | | |
| johnston-linear-matrix-algebra_FO2856 | 311 | 1.000 | c_{n-1} | | |
| johnston-linear-matrix-algebra_FO2857 | 311 | 1.000 | (-1)^{n-1}\left(\lambda_{1}+\lambda_{2}+\cdots+\lambda_{n}\right) | | |
| johnston-linear-matrix-algebra_FO2858 | 311 | 1.000 | (-1)^{n-1} c_{n-1}=\operatorname{tr}(A) | | |
| johnston-linear-matrix-algebra_FO2859 | 311 | 0.997 | n-1 | | |
| johnston-linear-matrix-algebra_FO2861 | 313 | 1.000 | A \in \mathcal{M}_{m, n}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO2862 | 313 | 0.742 | \lambda=1,1 | | |
| johnston-linear-matrix-algebra_FO2863 | 313 | 0.742 | \operatorname{tr}(A)= | | |
| johnston-linear-matrix-algebra_FO2864 | 313 | 1.000 | 1+1+2=4 | | |
| johnston-linear-matrix-algebra_FO2865 | 313 | 1.000 | \operatorname{det}(A)=1 \times 1 \times 2=2 | | |
| johnston-linear-matrix-algebra_FO2866 | 313 | 1.000 | A^{*} | | |
| johnston-linear-matrix-algebra_FO2867 | 313 | 1.000 | \left[\begin{array}{cc}2 i & 1-3 i \\ -3 & 5+2 i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2868 | 313 | 1.000 | \left[\begin{array}{cc}0 & 2+3 i \\ 2-3 i & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2869 | 313 | 1.000 | \left[\begin{array}{cc}i & 1-i \\ -2 & -2-i \\ 1+i & 2+3 i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2870 | 314 | 1.000 | \mathbf{v} \cdot(A \mathbf{w})=\left(A^{*} \mathbf{v}\right) \cdot \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO2871 | 314 | 1.000 | \mathbf{v}^{*} A \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2872 | 314 | 1.000 | \bar{z} z=|z|^{2} | | |
| johnston-linear-matrix-algebra_FO2873 | 314 | 1.000 | z \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO2874 | 314 | 1.000 | A^{*}=A | | |
| johnston-linear-matrix-algebra_FO2875 | 314 | 1.000 | \mathbf{v} \in \mathbb{C}^{n} | | |
| johnston-linear-matrix-algebra_FO2876 | 314 | 1.000 | \lambda \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO2877 | 315 | 1.000 | \lambda \mathbf{v}^{*} \mathbf{v}=\bar{\lambda} \mathbf{v}^{*} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2878 | 315 | 1.000 | \mathbf{v}^{*} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2879 | 315 | 1.000 | \lambda=\bar{\lambda} | | |
| johnston-linear-matrix-algebra_FO2880 | 315 | 1.000 | \lambda \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2881 | 315 | 1.000 | \left[\begin{array}{cc}2 & 1+i \\ 1-i & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2882 | 315 | 1.000 | \left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & -3 & 2 \\ -1 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2883 | 315 | 1.000 | -\lambda^{3}-2 \lambda^{2}+9 \lambda-2 | | |
| johnston-linear-matrix-algebra_FO2884 | 315 | 1.000 | \lambda-2 | | |
| johnston-linear-matrix-algebra_FO2885 | 315 | 0.972 | -\lambda^{3}-2 \lambda^{2}+9 \lambda-2=(\lambda-2)\left(-\lambda^{2}-4 \lambda+1\right) | | |
| johnston-linear-matrix-algebra_FO2886 | 315 | 1.000 | -\lambda^{2}-4 \lambda+1 | | |
| johnston-linear-matrix-algebra_FO2887 | 316 | 1.000 | \operatorname{null}(A-\lambda I) | | |
| johnston-linear-matrix-algebra_FO2888 | 316 | 0.943 | (-1,1,0),(0,1,-1) | | |
| johnston-linear-matrix-algebra_FO2889 | 316 | 0.943 | (1,1,0) | | |
| johnston-linear-matrix-algebra_FO2890 | 316 | 1.000 | \{(-1,1,0)\},\{(0,1,-1)\} | | |
| johnston-linear-matrix-algebra_FO2891 | 316 | 1.000 | \{(1,1,0)\} | | |
| johnston-linear-matrix-algebra_FO2892 | 316 | 1.000 | A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 1 & -1 & -1 \\ 0 & 0 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2893 | 317 | 0.925 | (A-\lambda I) | | |
| johnston-linear-matrix-algebra_FO2894 | 317 | 0.999 | (A-2 I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2895 | 317 | 1.000 | v_{1}=3 v_{2} | | |
| johnston-linear-matrix-algebra_FO2896 | 317 | 1.000 | v_{2}(3,1,0) | | |
| johnston-linear-matrix-algebra_FO2897 | 317 | 1.000 | \{(3,1,0)\} | | |
| johnston-linear-matrix-algebra_FO2898 | 317 | 1.000 | v_{1}=v_{3} | | |
| johnston-linear-matrix-algebra_FO2899 | 317 | 1.000 | v_{2}(0,1,0)+v_{3}(1,0,1) | | |
| johnston-linear-matrix-algebra_FO2900 | 317 | 1.000 | \{(0,1,0),(1,0,1)\} | | |
| johnston-linear-matrix-algebra_FO2901 | 318 | 1.000 | (B-\lambda I) | | |
| johnston-linear-matrix-algebra_FO2902 | 318 | 1.000 | B=\left[\begin{array}{ccc}2 & 2 & 3 \\ 0 & -1 & 0 \\ 0 & 1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2903 | 318 | 1.000 | (B+I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2904 | 318 | 1.000 | v_{1}=-v_{3} | | |
| johnston-linear-matrix-algebra_FO2905 | 318 | 1.000 | v_{2}=0 | | |
| johnston-linear-matrix-algebra_FO2906 | 318 | 1.000 | v_{3}(-1,0,1) | | |
| johnston-linear-matrix-algebra_FO2907 | 318 | 1.000 | \{(-1,0,1)\} | | |
| johnston-linear-matrix-algebra_FO2908 | 319 | 1.000 | \lambda_{1} | | |
| johnston-linear-matrix-algebra_FO2909 | 319 | 0.999 | \left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO2910 | 319 | 0.999 | \operatorname{nullity}\left(A-\lambda_{1} I\right) | | |
| johnston-linear-matrix-algebra_FO2911 | 319 | 1.000 | V \in \mathcal{M}_{n, n-k} | | |
| johnston-linear-matrix-algebra_FO2912 | 319 | 1.000 | B=\mathcal{M}_{k, n-k} | | |
| johnston-linear-matrix-algebra_FO2913 | 319 | 1.000 | C \in \mathcal{M}_{n-k, n-k} | | |
| johnston-linear-matrix-algebra_FO2914 | 319 | 1.000 | \left(\lambda_{1}-\lambda\right)^{k} | | |
| johnston-linear-matrix-algebra_FO2915 | 320 | 0.614 | \{(0,0,1,2)\} | | |
| johnston-linear-matrix-algebra_FO2916 | 320 | 1.000 | \lambda=-3 | | |
| johnston-linear-matrix-algebra_FO2917 | 320 | 1.000 | (A+3 I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO2918 | 320 | 1.000 | \{(0,1,0,0)\} | | |
| johnston-linear-matrix-algebra_FO2919 | 321 | 0.997 | \left.a_{2,2}, \ldots, a_{n, n}\right) | | |
| johnston-linear-matrix-algebra_FO2920 | 321 | 0.996 | \left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2921 | 321 | 1.000 | \left[\begin{array}{ccc}7 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2922 | 321 | 1.000 | \left[\begin{array}{lll}2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2923 | 322 | 0.852 | \{(1,0,0)\} | | |
| johnston-linear-matrix-algebra_FO2924 | 322 | 1.000 | A-\lambda I= | | |
| johnston-linear-matrix-algebra_FO2925 | 322 | 0.954 | \{(2,3,0)\} | | |
| johnston-linear-matrix-algebra_FO2926 | 322 | 1.000 | A-6 I | | |
| johnston-linear-matrix-algebra_FO2927 | 322 | 0.813 | \{(16,25,10)\} | | |
| johnston-linear-matrix-algebra_FO2928 | 322 | 0.938 | [\mathrm{v}],\left\{\mathbf{e}_{1}\right\},\left\{\mathbf{e}_{2}\right\} | | |
| johnston-linear-matrix-algebra_FO2929 | 322 | 1.000 | \left\{\mathbf{e}_{3}\right\} | | |
| johnston-linear-matrix-algebra_FO2930 | 322 | 0.957 | A=\left[\begin{array}{ll}1 & 5 \\ 4 & 2\end{array}\right], \mathbf{v}=\left[\begin{array}{l}1 \\ 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2931 | 322 | 1.000 | A=\left[\begin{array}{cc}1 & -1 \\ 3 & 6\end{array}\right], \mathbf{v}=\left[\begin{array}{c}-2 \\ 5+\sqrt{13}\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2932 | 322 | 0.921 | A=\left[\begin{array}{ccc}4 & 1 & 1 \\ 3 & 3 & -1 \\ 4 & 1 & 1\end{array}\right], \mathbf{v}=\left[\begin{array}{c}-4 \\ 7 \\ 9\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2933 | 322 | 1.000 | A=\left[\begin{array}{ccc}2 & -1 & -1 \\ -1 & 2 & 0 \\ 3 & 2 & 4\end{array}\right], \mathbf{v}=\left[\begin{array}{c}2 \\ -1+i \\ -1-3 i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2934 | 323 | 0.884 | A=\left[\begin{array}{llll}3 & 1 & 5 & 5 \\ 3 & 2 & 2 & 5 \\ 3 & 3 & 5 & 5 \\ 2 & 5 & 2 & 0\end{array}\right], \mathbf{v}=\left[\begin{array}{c}13 \\ 14 \\ 6 \\ -27\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2935 | 323 | 0.754 | \lambda=-3, A=\left[\begin{array}{ll}1 & 5 \\ 4 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2936 | 323 | 1.000 | \lambda=\sqrt{2}, A=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2937 | 323 | 0.985 | \lambda=2, A=\left[\begin{array}{ccc}0 & 3 & -1 \\ 2 & -1 & -1 \\ -2 & 3 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2938 | 323 | 1.000 | \lambda=2+i, A=\left[\begin{array}{ccc}-1 & 0 & -1 \\ 2 & 2 & 1 \\ 2 & -2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2939 | 323 | 0.969 | \lambda=-1, A=\left[\begin{array}{cccc}3 & 2 & 3 & 2 \\ 0 & 0 & 3 & -1 \\ 4 & 1 & 3 & 3 \\ 2 & 1 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2940 | 323 | 0.976 | \left[\begin{array}{cc}1 & 2 \\ -1 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2941 | 323 | 0.916 | \left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2942 | 323 | 1.000 | \left[\begin{array}{cc}2 & 1 \\ 3 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2943 | 323 | 0.599 | \left[\begin{array}{ccc}3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 7\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2944 | 323 | 0.992 | \left[\begin{array}{lll}2 & 3 & 0 \\ 3 & 0 & 1 \\ 0 & 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2945 | 323 | 0.981 | \left[\begin{array}{cccc}2 & 1 & 0 & 0 \\ 0 & -3 & 2 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2946 | 323 | 1.000 | \left[\begin{array}{cccc}2 & 1 & -1 & -1 \\ 0 & 3 & 1 & 1 \\ 0 & 1 & 3 & 1 \\ 0 & 0 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2947 | 323 | 0.999 | \left[\begin{array}{cccc}19 & 10 & 5 & 22 \\ 9 & 17 & 5 & 19 \\ -8 & -10 & 2 & -18 \\ -11 & -10 & -5 & -14\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2948 | 323 | 1.000 | \left[\begin{array}{cccc}12 & -3 & -6 & -3 \\ 1 & 16 & 2 & 1 \\ -3 & -3 & 9 & -3 \\ -1 & -1 & -2 & 14\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2949 | 323 | 1.000 | \left[\begin{array}{ccccc}6 & -4 & 4 & 0 & 4 \\ 0 & 10 & -6 & 0 & -6 \\ 16 & 20 & -20 & -8 & -26 \\ -8 & -8 & 11 & 10 & 11 \\ -16 & -20 & 24 & 8 & 30\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2950 | 323 | 1.000 | \operatorname{det}(A-\lambda I)=\operatorname{det}(\lambda I-A) | | |
| johnston-linear-matrix-algebra_FO2951 | 323 | 0.991 | A \in \mathcal{M}_{n}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO2952 | 323 | 1.000 | k \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2953 | 323 | 0.999 | A=\left[\begin{array}{ccc}1 & 0 & 0 \\ -8 & 4 & -5 \\ 8 & 0 & 9\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO2954 | 323 | 1.000 | A^{2}-3 A+I | | |
| johnston-linear-matrix-algebra_FO2955 | 323 | 0.998 | A^{2}- | | |
| johnston-linear-matrix-algebra_FO2956 | 323 | 1.000 | 3 A+I | | |
| johnston-linear-matrix-algebra_FO2957 | 323 | 1.000 | \overline{\mathbf{v}} | | |
| johnston-linear-matrix-algebra_FO2958 | 323 | 1.000 | \bar{\lambda} | | |
| johnston-linear-matrix-algebra_FO2959 | 324 | 0.928 | A \in \mathcal{M}_{2}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO2960 | 324 | 1.000 | A^{4}=I | | |
| johnston-linear-matrix-algebra_FO2961 | 324 | 1.000 | A^{*}=-A | | |
| johnston-linear-matrix-algebra_FO2962 | 324 | 0.991 | b \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO2963 | 324 | 0.955 | \mathbf{v}^{*} \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO2964 | 324 | 1.000 | \mathbf{v} \in \mathbb{C}^{m} | | |
| johnston-linear-matrix-algebra_FO2965 | 324 | 1.000 | \mathbf{w} \in \mathbb{C}^{n} | | |
| johnston-linear-matrix-algebra_FO2966 | 324 | 1.000 | B \in \mathcal{M}_{n, m}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO2967 | 324 | 1.000 | \mathbf{v} \cdot(A \mathbf{w})= | | |
| johnston-linear-matrix-algebra_FO2968 | 324 | 1.000 | (B \mathbf{v}) \cdot \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO2969 | 324 | 1.000 | B=A^{*} | | |
| johnston-linear-matrix-algebra_FO2970 | 324 | 1.000 | P \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2971 | 324 | 1.000 | \lambda \neq 0 | | |
| johnston-linear-matrix-algebra_FO2972 | 324 | 0.999 | B A B \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO2973 | 324 | 1.000 | \mu \neq \bar{\lambda} | | |
| johnston-linear-matrix-algebra_FO2974 | 324 | 0.997 | \lambda_{1}, \lambda_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO2975 | 324 | 0.992 | \lambda_{n} | | |
| johnston-linear-matrix-algebra_FO2976 | 324 | 1.000 | \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} | | |
| johnston-linear-matrix-algebra_FO2977 | 324 | 1.000 | B=A-\lambda_{1} \mathbf{v}_{1} \mathbf{v}_{1}^{*} | | |
| johnston-linear-matrix-algebra_FO2978 | 324 | 1.000 | 0, \lambda_{2}, \lambda_{3}, \ldots, \lambda_{n} | | |
| johnston-linear-matrix-algebra_FO2979 | 324 | 1.000 | \lambda_{1} \neq \lambda_{2} | | |
| johnston-linear-matrix-algebra_FO2980 | 324 | 0.999 | \lambda_{1} \neq \lambda_{3}, \ldots, \lambda_{1} \neq \lambda_{n} | | |
| johnston-linear-matrix-algebra_FO2981 | 324 | 1.000 | (-1)^{n} \lambda^{n} | | |
| johnston-linear-matrix-algebra_FO2982 | 324 | 1.000 | p_{C}(\lambda)=-\left(\lambda^{3}+a_{2} \lambda^{2}+a_{1} \lambda+a_{0}\right) | | |
| johnston-linear-matrix-algebra_FO2983 | 324 | 1.000 | p_{C}(\lambda)=(-1)^{n}\left(\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{1} \lambda+a_{0}\right) | | |
| johnston-linear-matrix-algebra_FO2984 | 324 | 1.000 | p_{C} | | |
| johnston-linear-matrix-algebra_FO2985 | 325 | 0.987 | v_{i} | | |
| johnston-linear-matrix-algebra_FO2986 | 325 | 0.987 | v_{i}=1 | | |
| johnston-linear-matrix-algebra_FO2987 | 325 | 1.000 | A=P D P^{-1} | | |
| johnston-linear-matrix-algebra_FO2988 | 325 | 1.000 | D^{k} | | |
| johnston-linear-matrix-algebra_FO2989 | 325 | 1.000 | P^{-1} | | |
| johnston-linear-matrix-algebra_FO2990 | 326 | 0.999 | P D | | |
| johnston-linear-matrix-algebra_FO2991 | 326 | 1.000 | P, D \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO2992 | 326 | 1.000 | A P=P D | | |
| johnston-linear-matrix-algebra_FO2993 | 326 | 1.000 | A P | | |
| johnston-linear-matrix-algebra_FO2994 | 326 | 1.000 | A \mathbf{v}_{j}=d_{j, j} \mathbf{v}_{j} | | |
| johnston-linear-matrix-algebra_FO2995 | 326 | 1.000 | d_{1,1}, d_{2,2}, \ldots, d_{n, n} | | |
| johnston-linear-matrix-algebra_FO2996 | 326 | 1.000 | \lambda_{1}=-1 | | |
| johnston-linear-matrix-algebra_FO2997 | 326 | 1.000 | \lambda_{2}=6 | | |
| johnston-linear-matrix-algebra_FO2998 | 326 | 0.982 | \mathbf{v}_{2}=(2,5) | | |
| johnston-linear-matrix-algebra_FO2999 | 327 | 1.000 | \mathbf{v}_{2}=(0,1,0) | | |
| johnston-linear-matrix-algebra_FO3000 | 327 | 0.998 | \mathbf{v}_{3}=(1,0,1) | | |
| johnston-linear-matrix-algebra_FO3001 | 327 | 1.000 | \mathbf{v}_{2}=(1,0,1) | | |
| johnston-linear-matrix-algebra_FO3002 | 327 | 1.000 | \mathbf{v}_{3}=(0,1,0) | | |
| johnston-linear-matrix-algebra_FO3003 | 327 | 1.000 | P D P^{-1} | | |
| johnston-linear-matrix-algebra_FO3004 | 327 | 1.000 | B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}=\{(-1,1),(2,5)\} | | |
| johnston-linear-matrix-algebra_FO3005 | 327 | 1.000 | [T]_{B}=D | | |
| johnston-linear-matrix-algebra_FO3006 | 327 | 1.000 | \lambda_{1}=2 | | |
| johnston-linear-matrix-algebra_FO3007 | 327 | 1.000 | \lambda_{2}=\lambda_{3}=-1 | | |
| johnston-linear-matrix-algebra_FO3008 | 327 | 1.000 | \mathbf{v}_{1}=(3,1,0), \mathbf{v}_{2}=(0,1,0) | | |
| johnston-linear-matrix-algebra_FO3009 | 328 | 1.000 | B_{1}, B_{2}, \ldots, B_{m} | | |
| johnston-linear-matrix-algebra_FO3010 | 328 | 1.000 | \mathbf{v}_{1} \in \operatorname{span}\left(B_{1}\right), \mathbf{v}_{2} \in \operatorname{span}\left(B_{2}\right), \ldots, \mathbf{v}_{m} \in \operatorname{span}\left(B_{m}\right) | | |
| johnston-linear-matrix-algebra_FO3011 | 329 | 1.000 | \gamma_{j} | | |
| johnston-linear-matrix-algebra_FO3012 | 329 | 1.000 | B_{j} | | |
| johnston-linear-matrix-algebra_FO3013 | 329 | 1.000 | \lambda_{1}, \lambda_{2}, \ldots, \lambda_{m} | | |
| johnston-linear-matrix-algebra_FO3014 | 329 | 1.000 | 1 \leq \ell \leq m | | |
| johnston-linear-matrix-algebra_FO3015 | 329 | 1.000 | \left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{\ell}\right\} | | |
| johnston-linear-matrix-algebra_FO3016 | 329 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{\ell} | | |
| johnston-linear-matrix-algebra_FO3017 | 329 | 1.000 | \mathbf{v}_{1}+\mathbf{v}_{2}+\cdots+\mathbf{v}_{m}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3018 | 329 | 1.000 | \mathbf{v}_{1}=\mathbf{v}_{2}=\cdots=\mathbf{v}_{m}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3019 | 329 | 1.000 | k \leq \ell | | |
| johnston-linear-matrix-algebra_FO3020 | 329 | 0.934 | c_{1}, c_{2}, \ldots, c_{k-1} | | |
| johnston-linear-matrix-algebra_FO3021 | 329 | 1.000 | \lambda_{k} | | |
| johnston-linear-matrix-algebra_FO3022 | 329 | 1.000 | \left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k-1}\right\} | | |
| johnston-linear-matrix-algebra_FO3023 | 329 | 1.000 | c_{j}\left(\lambda_{j}-\lambda_{k}\right)=0 | | |
| johnston-linear-matrix-algebra_FO3024 | 329 | 1.000 | 1 \leq j \leq k-1 | | |
| johnston-linear-matrix-algebra_FO3025 | 329 | 1.000 | c_{1}=c_{2}=\cdots=c_{k-1}=0 | | |
| johnston-linear-matrix-algebra_FO3026 | 329 | 1.000 | \mathbf{v}_{k}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3027 | 329 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{m} | | |
| johnston-linear-matrix-algebra_FO3028 | 329 | 1.000 | \mathbf{v}_{m}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3029 | 329 | 0.958 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{m-1} | | |
| johnston-linear-matrix-algebra_FO3030 | 329 | 1.000 | B_{1} \cup B_{2} \cup \cdots \cup B_{m} | | |
| johnston-linear-matrix-algebra_FO3031 | 329 | 1.000 | B_{j}=\left\{\mathbf{v}_{j, 1}, \mathbf{v}_{j, 2}, \ldots, \mathbf{v}_{j, \gamma_{j}}\right\} | | |
| johnston-linear-matrix-algebra_FO3032 | 329 | 1.000 | 1 \leq j \leq m | | |
| johnston-linear-matrix-algebra_FO3033 | 329 | 1.000 | c_{j, 1}=c_{j, 2}=\cdots=c_{j, \gamma_{j}}=0 | | |
| johnston-linear-matrix-algebra_FO3034 | 330 | 1.000 | A=\left[\begin{array}{cc}3 & -1 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3035 | 330 | 1.000 | \mathbf{v}=v_{2}(1,1) | | |
| johnston-linear-matrix-algebra_FO3036 | 331 | 1.000 | D, P \in \mathcal{M}_{n}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3037 | 331 | 1.000 | D, P \in \mathcal{M}_{n}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO3038 | 331 | 1.000 | \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO3039 | 331 | 1.000 | B_{1}, B_{2}, \ldots, B_{m} \subseteq B | | |
| johnston-linear-matrix-algebra_FO3040 | 331 | 0.995 | B_{1}, B_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO3041 | 331 | 1.000 | B_{m} | | |
| johnston-linear-matrix-algebra_FO3042 | 331 | 0.999 | \lambda_{j} | | |
| johnston-linear-matrix-algebra_FO3043 | 331 | 0.999 | \left|B_{j}\right| | | |
| johnston-linear-matrix-algebra_FO3044 | 331 | 0.999 | n=|B|=\left|B_{1}\right|+\cdots+\left|B_{m}\right| | | |
| johnston-linear-matrix-algebra_FO3045 | 332 | 1.000 | \left|B_{1}\right|,\left|B_{2}\right|, \ldots,\left|B_{m}\right| | | |
| johnston-linear-matrix-algebra_FO3046 | 332 | 0.998 | \left|B_{1}\right|+\left|B_{2}\right|+\cdots+\left|B_{m}\right|=n | | |
| johnston-linear-matrix-algebra_FO3047 | 332 | 0.998 | B=B_{1} \cup B_{2} \cup \cdots \cup B_{m} | | |
| johnston-linear-matrix-algebra_FO3048 | 332 | 0.998 | |B|=n | | |
| johnston-linear-matrix-algebra_FO3049 | 332 | 0.979 | 1+2=3=n | | |
| johnston-linear-matrix-algebra_FO3050 | 332 | 0.793 | \operatorname{not} A=\left[\begin{array}{ccc}5 & 1 & -1 \\ 1 & 3 & -1 \\ 2 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3051 | 333 | 1.000 | a \lambda^{2}+b \lambda+c | | |
| johnston-linear-matrix-algebra_FO3052 | 333 | 0.997 | 1+ | | |
| johnston-linear-matrix-algebra_FO3053 | 333 | 0.997 | 1+\cdots+1=n | | |
| johnston-linear-matrix-algebra_FO3054 | 333 | 0.972 | \operatorname{not} A=\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3055 | 334 | 1.000 | 1+\varepsilon | | |
| johnston-linear-matrix-algebra_FO3056 | 334 | 1.000 | \varepsilon \notin\{0,4\} | | |
| johnston-linear-matrix-algebra_FO3057 | 335 | 1.000 | Q^{*} | | |
| johnston-linear-matrix-algebra_FO3058 | 335 | 1.000 | Q | | |
| johnston-linear-matrix-algebra_FO3059 | 335 | 1.000 | A^{500} | | |
| johnston-linear-matrix-algebra_FO3060 | 335 | 1.000 | k=500 | | |
| johnston-linear-matrix-algebra_FO3061 | 335 | 1.000 | P, Q, D \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO3062 | 335 | 0.999 | d_{1}, d_{2}, \ldots, d_{n} | | |
| johnston-linear-matrix-algebra_FO3063 | 336 | 1.000 | \mathbf{p}_{1} \mathbf{q}_{1}^{*} | | |
| johnston-linear-matrix-algebra_FO3064 | 336 | 1.000 | (-1)^{k} | | |
| johnston-linear-matrix-algebra_FO3065 | 336 | 1.000 | \mathbf{p}_{j} | | |
| johnston-linear-matrix-algebra_FO3066 | 336 | 1.000 | \mathbf{q}_{j}^{*} | | |
| johnston-linear-matrix-algebra_FO3067 | 336 | 1.000 | A^{k}=P D^{k} P^{-1} | | |
| johnston-linear-matrix-algebra_FO3068 | 336 | 0.996 | \mathbf{p}_{j} \mathbf{q}_{j}^{*} | | |
| johnston-linear-matrix-algebra_FO3069 | 337 | 1.000 | F_{3}=F_{2}+F_{1}=1+1=2 | | |
| johnston-linear-matrix-algebra_FO3070 | 337 | 0.999 | F_{4}=F_{3}+F_{2}=2+1=3 | | |
| johnston-linear-matrix-algebra_FO3071 | 337 | 1.000 | \phi=1.61803 \ldots | | |
| johnston-linear-matrix-algebra_FO3072 | 337 | 1.000 | \mathbf{p}_{1} \mathbf{q}_{1}^{*}, \mathbf{p}_{2} \mathbf{q}_{2}^{*}, \ldots, \mathbf{p}_{n} \mathbf{q}_{n}^{*} | | |
| johnston-linear-matrix-algebra_FO3073 | 337 | 0.988 | 0,1,1,2,3,5,8,13,21,34, \ldots | | |
| johnston-linear-matrix-algebra_FO3074 | 337 | 1.000 | F_{n} | | |
| johnston-linear-matrix-algebra_FO3075 | 337 | 1.000 | F_{n+1}=F_{n}+F_{n-1} | | |
| johnston-linear-matrix-algebra_FO3076 | 337 | 1.000 | \lambda^{2}-\lambda-1 | | |
| johnston-linear-matrix-algebra_FO3077 | 337 | 1.000 | \phi=(1+\sqrt{5}) / 2 | | |
| johnston-linear-matrix-algebra_FO3078 | 337 | 0.999 | 1-\phi | | |
| johnston-linear-matrix-algebra_FO3079 | 337 | 1.000 | \lambda=\phi | | |
| johnston-linear-matrix-algebra_FO3080 | 338 | 1.000 | 1-(1-\phi)(-\phi)= | | |
| johnston-linear-matrix-algebra_FO3081 | 338 | 0.628 | \phi^{2}+\phi+1=0 | | |
| johnston-linear-matrix-algebra_FO3082 | 338 | 1.000 | A^{n}=P D^{n} P^{-1} | | |
| johnston-linear-matrix-algebra_FO3083 | 338 | 0.935 | \left(F_{n}\right) | | |
| johnston-linear-matrix-algebra_FO3084 | 338 | 0.830 | \operatorname{null}(A-\phi I) | | |
| johnston-linear-matrix-algebra_FO3085 | 338 | 1.000 | \{(\phi, 1)\} | | |
| johnston-linear-matrix-algebra_FO3086 | 338 | 1.000 | \lambda= | | |
| johnston-linear-matrix-algebra_FO3087 | 338 | 0.950 | \operatorname{null}(A-(1-\phi) I) | | |
| johnston-linear-matrix-algebra_FO3088 | 338 | 0.997 | \{(1-\phi, 1)\} | | |
| johnston-linear-matrix-algebra_FO3089 | 339 | 0.986 | \lambda_{3}=\lambda_{4}=0 | | |
| johnston-linear-matrix-algebra_FO3090 | 339 | 1.000 | \left[P D^{k} P^{-1}\right]_{1,4} | | |
| johnston-linear-matrix-algebra_FO3091 | 339 | 1.000 | D^{k} P^{-1} | | |
| johnston-linear-matrix-algebra_FO3092 | 339 | 0.978 | A \in \mathcal{M}_{5} | | |
| johnston-linear-matrix-algebra_FO3093 | 339 | 1.000 | \left[A^{k}\right]_{1,4} | | |
| johnston-linear-matrix-algebra_FO3094 | 339 | 1.000 | A^{k}= | | |
| johnston-linear-matrix-algebra_FO3095 | 339 | 1.000 | P D^{k} P^{-1} | | |
| johnston-linear-matrix-algebra_FO3096 | 339 | 1.000 | \left[A^{k}\right]_{1,4}=\left[P D^{k} P^{-1}\right]_{1,4} | | |
| johnston-linear-matrix-algebra_FO3097 | 340 | 1.000 | r \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO3098 | 340 | 1.000 | P D^{k} | | |
| johnston-linear-matrix-algebra_FO3099 | 340 | 0.991 | D^{r} | | |
| johnston-linear-matrix-algebra_FO3100 | 340 | 1.000 | A^{r} | | |
| johnston-linear-matrix-algebra_FO3101 | 340 | 1.000 | (-1)^{1 / 2}=i | | |
| johnston-linear-matrix-algebra_FO3102 | 340 | 1.000 | A=\left[\begin{array}{cc}0 & 4 \\ -1 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3103 | 340 | 1.000 | A^{1 / 2} | | |
| johnston-linear-matrix-algebra_FO3104 | 340 | 1.000 | A^{\pi} | | |
| johnston-linear-matrix-algebra_FO3105 | 340 | 0.710 | (4,1) | | |
| johnston-linear-matrix-algebra_FO3106 | 341 | 1.000 | r=1 / 2 | | |
| johnston-linear-matrix-algebra_FO3107 | 341 | 1.000 | r=-1 | | |
| johnston-linear-matrix-algebra_FO3108 | 341 | 1.000 | r=\pi | | |
| johnston-linear-matrix-algebra_FO3109 | 342 | 1.000 | \lambda_{j}^{-1} | | |
| johnston-linear-matrix-algebra_FO3110 | 342 | 1.000 | B^{2}=A | | |
| johnston-linear-matrix-algebra_FO3111 | 342 | 1.000 | B^{k}=A | | |
| johnston-linear-matrix-algebra_FO3112 | 342 | 1.000 | B=A^{1 / k} | | |
| johnston-linear-matrix-algebra_FO3113 | 342 | 1.000 | A=\left[\begin{array}{cc}-2 & 6 \\ 3 & -5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3114 | 342 | 0.719 | (1,-1) | | |
| johnston-linear-matrix-algebra_FO3115 | 343 | 1.000 | A^{1 / 2}, B | | |
| johnston-linear-matrix-algebra_FO3116 | 343 | 1.000 | B^{3}=A | | |
| johnston-linear-matrix-algebra_FO3117 | 343 | 1.000 | -A^{1 / 2} | | |
| johnston-linear-matrix-algebra_FO3118 | 343 | 0.978 | -B | | |
| johnston-linear-matrix-algebra_FO3119 | 344 | 0.531 | F_{\mathbf{u}}^{2}=I | | |
| johnston-linear-matrix-algebra_FO3120 | 344 | 0.990 | \left[F_{\mathbf{u}}\right]=2 \mathbf{u u}^{T}-I | | |
| johnston-linear-matrix-algebra_FO3121 | 344 | 0.446 | 2 \mathbf{u u}^{T}-I | | |
| johnston-linear-matrix-algebra_FO3122 | 344 | 0.446 | \left(2 \mathbf{u} \mathbf{u}^{T}-I\right)^{2}=I | | |
| johnston-linear-matrix-algebra_FO3123 | 344 | 1.000 | k>1 | | |
| johnston-linear-matrix-algebra_FO3124 | 344 | 1.000 | k^{n} | | |
| johnston-linear-matrix-algebra_FO3125 | 344 | 1.000 | A^{r+s}=A^{r} A^{s} | | |
| johnston-linear-matrix-algebra_FO3126 | 344 | 1.000 | \left(A^{r}\right)^{s}= | | |
| johnston-linear-matrix-algebra_FO3127 | 344 | 1.000 | A^{r s} | | |
| johnston-linear-matrix-algebra_FO3128 | 345 | 1.000 | p(D) | | |
| johnston-linear-matrix-algebra_FO3129 | 345 | 1.000 | p(x)=c_{k} x^{k}+\cdots+c_{2} x^{2}+c_{1} x+c_{0} | | |
| johnston-linear-matrix-algebra_FO3130 | 345 | 1.000 | \left(P D P^{-1}\right)^{r}=P D^{r} P^{-1} | | |
| johnston-linear-matrix-algebra_FO3131 | 345 | 1.000 | p(\lambda) | | |
| johnston-linear-matrix-algebra_FO3132 | 345 | 0.979 | p(A) | | |
| johnston-linear-matrix-algebra_FO3133 | 345 | 1.000 | p(x)=x^{3}-3 x^{2}+2 x-4 | | |
| johnston-linear-matrix-algebra_FO3134 | 346 | 0.991 | f(A) | | |
| johnston-linear-matrix-algebra_FO3135 | 346 | 1.000 | f(D) | | |
| johnston-linear-matrix-algebra_FO3136 | 346 | 1.000 | e^{A} | | |
| johnston-linear-matrix-algebra_FO3137 | 346 | 1.000 | \sin (A) | | |
| johnston-linear-matrix-algebra_FO3138 | 347 | 1.000 | \sin (x) | | |
| johnston-linear-matrix-algebra_FO3139 | 347 | 1.000 | e^{x} | | |
| johnston-linear-matrix-algebra_FO3140 | 347 | 0.996 | f(\lambda) | | |
| johnston-linear-matrix-algebra_FO3141 | 347 | 1.000 | (a, b) | | |
| johnston-linear-matrix-algebra_FO3142 | 348 | 1.000 | f(A)=e^{A} | | |
| johnston-linear-matrix-algebra_FO3143 | 348 | 1.000 | e^{0}=1 | | |
| johnston-linear-matrix-algebra_FO3144 | 348 | 1.000 | 1 / e^{x}=e^{-x} | | |
| johnston-linear-matrix-algebra_FO3145 | 348 | 1.000 | e^{O}=I | | |
| johnston-linear-matrix-algebra_FO3146 | 348 | 1.000 | \left(e^{A}\right)^{-1}=e^{-A} | | |
| johnston-linear-matrix-algebra_FO3147 | 348 | 1.000 | e^{-A} | | |
| johnston-linear-matrix-algebra_FO3148 | 349 | 1.000 | e^{A+B}=e^{A} e^{B} | | |
| johnston-linear-matrix-algebra_FO3149 | 349 | 1.000 | e^{x+y}=e^{x} e^{y} | | |
| johnston-linear-matrix-algebra_FO3150 | 349 | 1.000 | \operatorname{det}\left(e^{A}\right)=e^{\operatorname{tr}(A)} | | |
| johnston-linear-matrix-algebra_FO3151 | 349 | 1.000 | e^{\lambda_{1}}, e^{\lambda_{2}}, \ldots, e^{\lambda_{n}} | | |
| johnston-linear-matrix-algebra_FO3152 | 350 | 0.214 | *(\mathbf{c})\left[\begin{array}{l}1 \\ 1 \\ 0\end{array} \quad 1\right] | | |
| johnston-linear-matrix-algebra_FO3153 | 350 | 0.643 | \begin{gathered}{\left[\begin{array}{cc}2 & 1 \\ -1 & 2\end{array}\right]} \\ \text { (d) }\left[\begin{array}{ll}-2 & 6 \\ -3 & 7\end{array}\right]\end{gathered} | | |
| johnston-linear-matrix-algebra_FO3154 | 350 | 0.395 | \left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3155 | 350 | 1.000 | \left[\begin{array}{ccc}5 & 0 & -3 \\ -3 & 2 & 3 \\ 6 & 0 & -4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3156 | 350 | 0.705 | \left[\begin{array}{ccc}3 & 0 & 1 \\ 0 & -1 & 2 \\ 0 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3157 | 350 | 1.000 | \left[\begin{array}{ccc}-3 & 4 & 5 \\ -3 & 5 & 3 \\ 1 & -2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3158 | 350 | 0.262 | \left.\begin{array}{c}*(\mathbf{a}) \\ *(\mathbf{c})\end{array} \begin{array}{cc}1 & 1 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}2 & 1 \\ -1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3159 | 350 | 1.000 | \left[\begin{array}{cc}2+2 i & 1-i \\ 2 & 1-i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3160 | 350 | 1.000 | \left[\begin{array}{cc}1 & 1 \\ -9 & -5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3161 | 350 | 0.741 | \left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3162 | 350 | 1.000 | \left[\begin{array}{ccc}-2 & -3 & -2 \\ 4 & 6 & 4 \\ -4 & -5 & -4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3163 | 350 | 0.932 | \left[\begin{array}{ccc}1+i & -1 & 0 \\ 1 & -1+i & 0 \\ 2 & -2 & i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3164 | 350 | 1.000 | \left[\begin{array}{ccc}2 & 4 & -2 \\ 1 & 3 & -1 \\ 3 & -1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3165 | 350 | 1.000 | \sqrt{A} | | |
| johnston-linear-matrix-algebra_FO3166 | 351 | 1.000 | L_{n} | | |
| johnston-linear-matrix-algebra_FO3167 | 351 | 1.000 | P_{n} | | |
| johnston-linear-matrix-algebra_FO3168 | 351 | 1.000 | F_{n+1} F_{n-1}-F_{n}^{2}=(-1)^{n} | | |
| johnston-linear-matrix-algebra_FO3169 | 351 | 1.000 | F_{m+n}=F_{m+1} F_{n}+F_{m} F_{n-1} | | |
| johnston-linear-matrix-algebra_FO3170 | 351 | 1.000 | m, n \geq 1 | | |
| johnston-linear-matrix-algebra_FO3171 | 351 | 0.996 | A^{m} A^{n}=A^{m+n} | | |
| johnston-linear-matrix-algebra_FO3172 | 351 | 1.000 | A=\mathbf{x y}^{T} | | |
| johnston-linear-matrix-algebra_FO3173 | 351 | 0.954 | \mathbf{x} \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO3174 | 351 | 1.000 | \mathbf{x} \cdot \mathbf{y} \neq 0 | | |
| johnston-linear-matrix-algebra_FO3175 | 351 | 1.000 | A \neq B | | |
| johnston-linear-matrix-algebra_FO3176 | 351 | 1.000 | P_{1} D_{1} P_{1}^{-1}=P_{2} D_{2} P_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO3177 | 351 | 1.000 | P_{1} D_{1}^{r} P_{1}^{-1}=P_{2} D_{2}^{r} P_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO3178 | 351 | 1.000 | P_{1} f\left(D_{1}\right) P_{1}^{-1}= | | |
| johnston-linear-matrix-algebra_FO3179 | 351 | 1.000 | P_{2} f\left(D_{2}\right) P_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO3180 | 351 | 1.000 | \lambda^{k} | | |
| johnston-linear-matrix-algebra_FO3181 | 351 | 1.000 | \left(A^{r}\right)^{s}=A^{r s} | | |
| johnston-linear-matrix-algebra_FO3182 | 351 | 0.999 | e^{\left(A^{T}\right)}=\left(e^{A}\right)^{T} | | |
| johnston-linear-matrix-algebra_FO3183 | 351 | 1.000 | e^{A+B}= | | |
| johnston-linear-matrix-algebra_FO3184 | 351 | 1.000 | e^{A} e^{B} | | |
| johnston-linear-matrix-algebra_FO3185 | 351 | 1.000 | \sin ^{2}(A)+\cos ^{2}(A)=I | | |
| johnston-linear-matrix-algebra_FO3186 | 351 | 1.000 | p(\lambda)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+ | | |
| johnston-linear-matrix-algebra_FO3187 | 351 | 1.000 | a_{1} \lambda+a_{0} | | |
| johnston-linear-matrix-algebra_FO3188 | 351 | 1.000 | C=V D V^{-1} | | |
| johnston-linear-matrix-algebra_FO3190 | 352 | 1.000 | \lambda_{2} | | |
| johnston-linear-matrix-algebra_FO3191 | 353 | 1.000 | N | | |
| johnston-linear-matrix-algebra_FO3192 | 354 | 1.000 | A, B \in \mathcal{M}_{n}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO3193 | 354 | 1.000 | B=Q D Q^{-1} | | |
| johnston-linear-matrix-algebra_FO3194 | 354 | 1.000 | Q^{-1} | | |
| johnston-linear-matrix-algebra_FO3195 | 354 | 1.000 | Q^{-1} B Q=D | | |
| johnston-linear-matrix-algebra_FO3196 | 354 | 0.983 | \left(P Q^{-1}\right)^{-1}=Q P^{-1} | | |
| johnston-linear-matrix-algebra_FO3197 | 354 | 0.999 | B=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3198 | 354 | 1.000 | B=\left[\begin{array}{ll}2 & 0 \\ 3 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3199 | 354 | 0.996 | A=\left[\begin{array}{ll}3 & 1 \\ 2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3200 | 354 | 0.996 | B=\left[\begin{array}{cc}2 & -1 \\ 3 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3201 | 354 | 1.000 | A=\left[\begin{array}{cc}2 & 1 \\ -3 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3202 | 354 | 1.000 | B=\left[\begin{array}{cc}4 & 2 \\ -1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3203 | 354 | 0.980 | A=\left[\begin{array}{ll}3 & 3 \\ 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3204 | 354 | 0.999 | A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3205 | 354 | 0.999 | B=\left[\begin{array}{lll}1 & 2 & 3 \\ 8 & 9 & 4 \\ 7 & 6 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3206 | 354 | 0.962 | A=\left[\begin{array}{ccc}3 & 1 & -1 \\ 0 & 2 & 1 \\ -1 & 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3207 | 354 | 0.962 | B=\left[\begin{array}{ccc}3 & 1 & 2 \\ -2 & 3 & 0 \\ 1 & -1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3208 | 354 | 0.736 | B \in \mathcal{M}_{3} | | |
| johnston-linear-matrix-algebra_FO3209 | 355 | 1.000 | A P=P B | | |
| johnston-linear-matrix-algebra_FO3210 | 355 | 1.000 | A^{2}=N | | |
| johnston-linear-matrix-algebra_FO3211 | 355 | 1.000 | c_{i, j}=(-1)^{i+j} m_{i, j} | | |
| johnston-linear-matrix-algebra_FO3212 | 355 | 0.993 | \operatorname{cof}(A) | | |
| johnston-linear-matrix-algebra_FO3213 | 355 | 1.000 | \left[\begin{array}{cc}2 & 3 \\ -1 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3214 | 356 | 1.000 | \operatorname{cof}(A)^{T} | | |
| johnston-linear-matrix-algebra_FO3215 | 356 | 0.998 | \operatorname{adj}(A) | | |
| johnston-linear-matrix-algebra_FO3216 | 356 | 0.980 | A \operatorname{cof}(A)^{T} | | |
| johnston-linear-matrix-algebra_FO3217 | 356 | 0.660 | \operatorname{product} A \operatorname{cof}(A)^{T} | | |
| johnston-linear-matrix-algebra_FO3218 | 357 | 1.000 | \left[A \operatorname{cof}(A)^{T}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO3219 | 357 | 1.000 | 0=\operatorname{det}(B)= | | |
| johnston-linear-matrix-algebra_FO3220 | 357 | 1.000 | A \operatorname{cof}(A)^{T}=\operatorname{det}(A) I | | |
| johnston-linear-matrix-algebra_FO3221 | 357 | 1.000 | \operatorname{det}(A)=a d-b c | | |
| johnston-linear-matrix-algebra_FO3222 | 358 | 1.000 | A_{j} | | |
| johnston-linear-matrix-algebra_FO3223 | 359 | 1.000 | A^{-1} \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO3224 | 359 | 1.000 | c_{1, j}, c_{2, j}, \ldots, c_{n, j} | | |
| johnston-linear-matrix-algebra_FO3225 | 359 | 1.000 | c_{1, j} b_{1}+c_{2, j} b_{2}+ | | |
| johnston-linear-matrix-algebra_FO3226 | 359 | 1.000 | \cdots+c_{n, j} b_{n} | | |
| johnston-linear-matrix-algebra_FO3227 | 359 | 1.000 | c_{1, j} b_{1}+ | | |
| johnston-linear-matrix-algebra_FO3228 | 359 | 1.000 | c_{2, j} b_{2}+\cdots+c_{n, j} b_{n}=\operatorname{det}\left(A_{j}\right) | | |
| johnston-linear-matrix-algebra_FO3229 | 359 | 1.000 | x_{j}=\operatorname{det}\left(A_{j}\right) / \operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO3230 | 359 | 1.000 | A, A_{1}, A_{2} | | |
| johnston-linear-matrix-algebra_FO3231 | 359 | 1.000 | A_{3} | | |
| johnston-linear-matrix-algebra_FO3232 | 361 | 1.000 | \sigma:\{1,2, \ldots, n\} \rightarrow | | |
| johnston-linear-matrix-algebra_FO3233 | 361 | 0.999 | \{1,2, \ldots, n\} | | |
| johnston-linear-matrix-algebra_FO3234 | 361 | 1.000 | \sigma | | |
| johnston-linear-matrix-algebra_FO3235 | 361 | 1.000 | 1,2, \ldots, n | | |
| johnston-linear-matrix-algebra_FO3236 | 361 | 1.000 | \sigma:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\} | | |
| johnston-linear-matrix-algebra_FO3237 | 361 | 0.996 | \sigma(i) \neq \sigma(j) | | |
| johnston-linear-matrix-algebra_FO3238 | 361 | 1.000 | \sigma:\{1,2,3\} \rightarrow\{1,2,3\} | | |
| johnston-linear-matrix-algebra_FO3239 | 361 | 1.000 | \sigma(1)=2 | | |
| johnston-linear-matrix-algebra_FO3240 | 361 | 0.973 | \sigma(2)=3 | | |
| johnston-linear-matrix-algebra_FO3241 | 361 | 0.973 | \sigma(3)=1 | | |
| johnston-linear-matrix-algebra_FO3242 | 361 | 1.000 | f:\{1,2,3\} \rightarrow\{1,2,3\} | | |
| johnston-linear-matrix-algebra_FO3243 | 361 | 1.000 | f(1)=2, f(2)=2 | | |
| johnston-linear-matrix-algebra_FO3244 | 361 | 1.000 | f(3)=1 | | |
| johnston-linear-matrix-algebra_FO3245 | 361 | 1.000 | f(1)=f(2) | | |
| johnston-linear-matrix-algebra_FO3246 | 361 | 1.000 | (\sigma(1) \sigma(2) \cdots \sigma(n)) | | |
| johnston-linear-matrix-algebra_FO3247 | 361 | 1.000 | \sigma(1)=2, \sigma(2)=3 | | |
| johnston-linear-matrix-algebra_FO3248 | 361 | 0.699 | \sigma=(4132) | | |
| johnston-linear-matrix-algebra_FO3249 | 361 | 1.000 | \sigma(1)=4, \sigma(2)=1, \sigma(3)=3 | | |
| johnston-linear-matrix-algebra_FO3250 | 361 | 1.000 | \sigma(4)=2 | | |
| johnston-linear-matrix-algebra_FO3251 | 361 | 1.000 | n!=n \cdot(n-1) \cdot(n-2) \cdots 2 \cdot 1 | | |
| johnston-linear-matrix-algebra_FO3252 | 361 | 1.000 | \sigma(1) | | |
| johnston-linear-matrix-algebra_FO3253 | 361 | 1.000 | \sigma(2) | | |
| johnston-linear-matrix-algebra_FO3254 | 361 | 0.970 | \sigma(3) | | |
| johnston-linear-matrix-algebra_FO3255 | 361 | 0.970 | n-2 | | |
| johnston-linear-matrix-algebra_FO3256 | 361 | 1.000 | S_{n} | | |
| johnston-linear-matrix-algebra_FO3257 | 361 | 1.000 | S_{2} | | |
| johnston-linear-matrix-algebra_FO3258 | 361 | 1.000 | S_{3} | | |
| johnston-linear-matrix-algebra_FO3259 | 361 | 0.772 | 2!=2 | | |
| johnston-linear-matrix-algebra_FO3260 | 361 | 0.772 | \{1,2\} | | |
| johnston-linear-matrix-algebra_FO3261 | 361 | 0.984 | 3!=6 | | |
| johnston-linear-matrix-algebra_FO3262 | 361 | 0.984 | \{1,2,3\} | | |
| johnston-linear-matrix-algebra_FO3263 | 361 | 1.000 | \sigma, \tau \in S_{n} | | |
| johnston-linear-matrix-algebra_FO3264 | 361 | 1.000 | \sigma \circ \tau | | |
| johnston-linear-matrix-algebra_FO3265 | 361 | 0.600 | \iota | | |
| johnston-linear-matrix-algebra_FO3266 | 361 | 0.644 | \imath=(12 \cdots n) | | |
| johnston-linear-matrix-algebra_FO3267 | 361 | 0.644 | \imath(j)=j | | |
| johnston-linear-matrix-algebra_FO3268 | 361 | 1.000 | \sigma^{-1} | | |
| johnston-linear-matrix-algebra_FO3269 | 362 | 1.000 | \sigma(\tau(1)) | | |
| johnston-linear-matrix-algebra_FO3270 | 362 | 1.000 | \sigma(\tau(2)), \ldots, \sigma(\tau(5)) | | |
| johnston-linear-matrix-algebra_FO3271 | 362 | 1.000 | \sigma(j)=k | | |
| johnston-linear-matrix-algebra_FO3272 | 362 | 1.000 | \sigma^{-1}(k)=j | | |
| johnston-linear-matrix-algebra_FO3273 | 362 | 0.306 | \sigma=(32514) | | |
| johnston-linear-matrix-algebra_FO3274 | 362 | 0.306 | \tau=(5142 | | |
| johnston-linear-matrix-algebra_FO3275 | 362 | 1.000 | \tau \circ \sigma | | |
| johnston-linear-matrix-algebra_FO3276 | 362 | 1.000 | \tau^{-1} | | |
| johnston-linear-matrix-algebra_FO3277 | 362 | 1.000 | \sigma(\tau(j)) | | |
| johnston-linear-matrix-algebra_FO3278 | 362 | 1.000 | 1 \leq j \leq 5 | | |
| johnston-linear-matrix-algebra_FO3279 | 362 | 1.000 | \sigma \circ \tau=(43125) | | |
| johnston-linear-matrix-algebra_FO3280 | 362 | 1.000 | \tau(\sigma(j)) | | |
| johnston-linear-matrix-algebra_FO3281 | 362 | 0.999 | \tau \circ \sigma=(41352) | | |
| johnston-linear-matrix-algebra_FO3282 | 362 | 0.994 | \sigma^{-1}=(42153) | | |
| johnston-linear-matrix-algebra_FO3283 | 362 | 0.637 | \tau | | |
| johnston-linear-matrix-algebra_FO3284 | 362 | 1.000 | \tau^{-1}=(24531) | | |
| johnston-linear-matrix-algebra_FO3285 | 362 | 1.000 | P_{\sigma} | | |
| johnston-linear-matrix-algebra_FO3286 | 362 | 1.000 | \mathbf{e}_{\sigma(j)} | | |
| johnston-linear-matrix-algebra_FO3287 | 362 | 1.000 | \sigma=(3421) | | |
| johnston-linear-matrix-algebra_FO3288 | 362 | 1.000 | \sigma=(25134) | | |
| johnston-linear-matrix-algebra_FO3289 | 362 | 0.956 | \imath \in S_{6} | | |
| johnston-linear-matrix-algebra_FO3292 | 363 | 1.000 | \mathbf{e}_{3}, \mathbf{e}_{4}, \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO3293 | 363 | 1.000 | \mathbf{e}_{2}, \mathbf{e}_{5}, \mathbf{e}_{1}, \mathbf{e}_{3} | | |
| johnston-linear-matrix-algebra_FO3294 | 363 | 1.000 | \mathbf{e}_{4} | | |
| johnston-linear-matrix-algebra_FO3295 | 363 | 1.000 | \{1,2,3,4,5,6\} | | |
| johnston-linear-matrix-algebra_FO3296 | 363 | 1.000 | \mathbf{e}_{1}, \mathbf{e}_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO3297 | 363 | 0.969 | \mathbf{e}_{6} | | |
| johnston-linear-matrix-algebra_FO3298 | 363 | 0.969 | 6 \times 6 | | |
| johnston-linear-matrix-algebra_FO3299 | 363 | 0.999 | S_{6} | | |
| johnston-linear-matrix-algebra_FO3300 | 363 | 0.999 | \mathcal{M}_{6} | | |
| johnston-linear-matrix-algebra_FO3301 | 363 | 1.000 | P_{\sigma}, P_{\tau} \in \mathcal{M}_{n} | | |
| johnston-linear-matrix-algebra_FO3302 | 363 | 1.000 | P_{\sigma} P_{\tau}=P_{\sigma \circ \tau} | | |
| johnston-linear-matrix-algebra_FO3303 | 363 | 1.000 | P_{\sigma}^{-1}=P_{\sigma}^{T}=P_{\sigma^{-1}} | | |
| johnston-linear-matrix-algebra_FO3304 | 364 | 1.000 | P_{\sigma \circ \tau} | | |
| johnston-linear-matrix-algebra_FO3305 | 364 | 0.843 | \mathbf{e}_{(\sigma \circ \tau)(j)} | | |
| johnston-linear-matrix-algebra_FO3306 | 364 | 1.000 | P_{\sigma} P_{\tau} | | |
| johnston-linear-matrix-algebra_FO3307 | 364 | 0.618 | P_{\sigma} P_{\sigma^{-1}}=P_{\sigma \circ \sigma^{-1}}=P_{\imath}=I | | |
| johnston-linear-matrix-algebra_FO3308 | 364 | 1.000 | P_{\sigma}^{-1}=P_{\sigma^{-1}} | | |
| johnston-linear-matrix-algebra_FO3309 | 364 | 1.000 | P_{\sigma}^{T}=P_{\sigma}^{-1} | | |
| johnston-linear-matrix-algebra_FO3310 | 364 | 1.000 | P_{\sigma}^{T} P_{\sigma} | | |
| johnston-linear-matrix-algebra_FO3311 | 364 | 1.000 | \sigma(i)=\sigma(j) | | |
| johnston-linear-matrix-algebra_FO3312 | 364 | 0.998 | \mathbf{e}_{\sigma(i)} \cdot \mathbf{e}_{\sigma(j)}=0 | | |
| johnston-linear-matrix-algebra_FO3313 | 364 | 1.000 | P_{\sigma}^{T} P_{\sigma}=I | | |
| johnston-linear-matrix-algebra_FO3314 | 364 | 1.000 | \operatorname{sgn}(\sigma) | | |
| johnston-linear-matrix-algebra_FO3315 | 364 | 0.998 | \sigma=\left(\begin{array}{llll}3 & 4 & 2 & 1\end{array}\right) | | |
| johnston-linear-matrix-algebra_FO3316 | 364 | 0.804 | \sigma=\left(\begin{array}{lllll}2 & 5 & 1 & 3 & 4\end{array}\right) | | |
| johnston-linear-matrix-algebra_FO3317 | 365 | 1.000 | (-1)^{s} | | |
| johnston-linear-matrix-algebra_FO3318 | 365 | 1.000 | \operatorname{sgn}(\sigma)=(-1)^{3}=-1 | | |
| johnston-linear-matrix-algebra_FO3319 | 365 | 1.000 | \operatorname{sgn}(\sigma)=(-1)^{4}=1 | | |
| johnston-linear-matrix-algebra_FO3320 | 365 | 0.936 | \operatorname{det}(A)=\sum_{\sigma \in S_{n}} \operatorname{sgn}(\sigma) a_{\sigma(1), 1} a_{\sigma(2), 2} \cdots a_{\sigma(n), n} | | |
| johnston-linear-matrix-algebra_FO3321 | 365 | 0.860 | \operatorname{det}(A)=a_{1,1} a_{2,2}-a_{2,1} a_{1,2} | | |
| johnston-linear-matrix-algebra_FO3322 | 365 | 0.998 | n=3 | | |
| johnston-linear-matrix-algebra_FO3329 | 366 | 1.000 | \mathbf{a}_{j} | | |
| johnston-linear-matrix-algebra_FO3330 | 366 | 1.000 | i_{1}, i_{2}, \ldots, i_{n} | | |
| johnston-linear-matrix-algebra_FO3331 | 366 | 1.000 | \left[\mathbf{e}_{i_{1}}\left|\mathbf{e}_{i_{2}}\right| \cdots \mid \mathbf{e}_{i_{n}}\right] | | |
| johnston-linear-matrix-algebra_FO3332 | 366 | 1.000 | \sigma(1)=i_{1}, \sigma(2)=i_{2} | | |
| johnston-linear-matrix-algebra_FO3333 | 367 | 1.000 | [A B]_{i, j} | | |
| johnston-linear-matrix-algebra_FO3334 | 367 | 0.996 | \operatorname{det}\left(A_{j_{1}, \ldots, j_{m}}\right) | | |
| johnston-linear-matrix-algebra_FO3335 | 367 | 0.887 | A_{j_{1}, \ldots, j_{m}} | | |
| johnston-linear-matrix-algebra_FO3336 | 367 | 1.000 | B_{j_{1}, \ldots, j_{m}} | | |
| johnston-linear-matrix-algebra_FO3337 | 367 | 1.000 | m \times m | | |
| johnston-linear-matrix-algebra_FO3338 | 367 | 1.000 | j_{1}, \ldots, j_{m} | | |
| johnston-linear-matrix-algebra_FO3339 | 367 | 1.000 | \left(j_{1}, j_{2}, \ldots, j_{m}\right) | | |
| johnston-linear-matrix-algebra_FO3340 | 367 | 1.000 | j_{1} \leq j_{2} \leq \cdots \leq j_{m} | | |
| johnston-linear-matrix-algebra_FO3341 | 367 | 1.000 | \operatorname{det}\left(A_{j_{1}, \ldots, j_{m}}\right)=0 | | |
| johnston-linear-matrix-algebra_FO3342 | 367 | 1.000 | 1 \leq j_{1} \leq \cdots \leq j_{m} \leq n | | |
| johnston-linear-matrix-algebra_FO3343 | 367 | 1.000 | 1 \leq j_{1}< | | |
| johnston-linear-matrix-algebra_FO3344 | 367 | 1.000 | \cdots<j_{m} \leq n | | |
| johnston-linear-matrix-algebra_FO3345 | 367 | 0.813 | \left(j_{1}, j_{2}, \ldots, j_{n}\right) | | |
| johnston-linear-matrix-algebra_FO3346 | 367 | 1.000 | 1 \leq j_{1}<j_{2}<\cdots<j_{n} \leq n | | |
| johnston-linear-matrix-algebra_FO3347 | 367 | 1.000 | \left(j_{1}, j_{2}, \ldots, j_{n}\right)=(1,2, \ldots, n) | | |
| johnston-linear-matrix-algebra_FO3348 | 368 | 1.000 | m=1 | | |
| johnston-linear-matrix-algebra_FO3349 | 368 | 1.000 | A \in \mathcal{M}_{1, n} | | |
| johnston-linear-matrix-algebra_FO3350 | 368 | 1.000 | B \in \mathcal{M}_{n, 1} | | |
| johnston-linear-matrix-algebra_FO3351 | 368 | 0.998 | \left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 2 & 3 \\ 4 & 5 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3352 | 369 | 0.995 | \left[\begin{array}{llll}3 & 2 & 4 & 3 \\ 1 & 1 & 3 & 2 \\ 4 & 0 & 1 & 1 \\ 3 & 1 & 2 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3353 | 369 | 1.000 | \left[\begin{array}{lllll}6 & 2 & 1 & 0 & 4 \\ 4 & 2 & 0 & 4 & 6 \\ 0 & 6 & 3 & 4 & 4 \\ 0 & 3 & 6 & 0 & 3 \\ 2 & 6 & 1 & 6 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3354 | 369 | 0.264 | \circ | | |
| johnston-linear-matrix-algebra_FO3355 | 369 | 0.087 | \circ(2331) | | |
| johnston-linear-matrix-algebra_FO3356 | 369 | 0.079 | \circ(42.331) | | |
| johnston-linear-matrix-algebra_FO3357 | 369 | 0.649 | (24113) \circ(4132) | | |
| johnston-linear-matrix-algebra_FO3358 | 369 | 0.792 | (241536) \circ(361524) | | |
| johnston-linear-matrix-algebra_FO3359 | 369 | 0.999 | \operatorname{cof}(I)=I | | |
| johnston-linear-matrix-algebra_FO3360 | 369 | 1.000 | \operatorname{cof}(A)=\operatorname{cof}(B) | | |
| johnston-linear-matrix-algebra_FO3361 | 369 | 1.000 | \operatorname{rank}(\operatorname{cof}(A)) | | |
| johnston-linear-matrix-algebra_FO3362 | 369 | 0.998 | \imath^{-1}=\imath | | |
| johnston-linear-matrix-algebra_FO3363 | 369 | 1.000 | \sigma \circ | | |
| johnston-linear-matrix-algebra_FO3364 | 369 | 1.000 | \tau=\tau \circ \sigma | | |
| johnston-linear-matrix-algebra_FO3365 | 369 | 1.000 | S_{5} | | |
| johnston-linear-matrix-algebra_FO3366 | 369 | 0.998 | \operatorname{cof}(A B)=\operatorname{cof}(A) \operatorname{cof}(B) | | |
| johnston-linear-matrix-algebra_FO3367 | 369 | 1.000 | \operatorname{cof}\left(A^{k}\right)=(\operatorname{cof}(A))^{k} | | |
| johnston-linear-matrix-algebra_FO3368 | 369 | 1.000 | \operatorname{cof}(c A)= | | |
| johnston-linear-matrix-algebra_FO3369 | 369 | 1.000 | c^{n-1} \operatorname{cof}(A) | | |
| johnston-linear-matrix-algebra_FO3370 | 369 | 1.000 | \operatorname{det}(\operatorname{cof}(A))=(\operatorname{det}(A))^{n-1} | | |
| johnston-linear-matrix-algebra_FO3371 | 369 | 1.000 | \operatorname{cof}(\operatorname{cof}(A))=(\operatorname{det}(A))^{n-2} A | | |
| johnston-linear-matrix-algebra_FO3372 | 369 | 1.000 | \operatorname{cof}(\operatorname{cof}(A))=O | | |
| johnston-linear-matrix-algebra_FO3373 | 369 | 1.000 | A \in \mathcal{M}_{n}, A \mathbf{x}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO3374 | 369 | 1.000 | A^{-1} A_{j} | | |
| johnston-linear-matrix-algebra_FO3375 | 369 | 1.000 | \operatorname{det}\left(A^{-1} A_{j}\right) | | |
| johnston-linear-matrix-algebra_FO3376 | 369 | 1.000 | \operatorname{sgn}(\sigma \circ \tau)=\operatorname{sgn}(\sigma) \cdot \operatorname{sgn}(\tau) | | |
| johnston-linear-matrix-algebra_FO3377 | 369 | 1.000 | m>n | | |
| johnston-linear-matrix-algebra_FO3378 | 370 | 1.000 | \operatorname{per}(A) | | |
| johnston-linear-matrix-algebra_FO3379 | 370 | 1.000 | \left[\begin{array}{cc}1 & 3 \\ 0 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3380 | 370 | 1.000 | \left[\begin{array}{ccc}1 & 1 & 3 \\ -4 & 2 & 1 \\ 3 & 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3381 | 370 | 1.000 | \operatorname{per}\left(A^{T}\right)=\operatorname{per}(A) | | |
| johnston-linear-matrix-algebra_FO3382 | 370 | 1.000 | \operatorname{per}(A)=a_{1,1} a_{2,2} \cdots a_{n, n} | | |
| johnston-linear-matrix-algebra_FO3383 | 370 | 0.999 | \operatorname{per}(A B)= | | |
| johnston-linear-matrix-algebra_FO3384 | 370 | 1.000 | \operatorname{per}(A) \operatorname{per}(B) | | |
| johnston-linear-matrix-algebra_FO3385 | 370 | 1.000 | \mathbf{v}_{0} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO3388 | 371 | 1.000 | \mathbf{v}_{k-1} | | |
| johnston-linear-matrix-algebra_FO3389 | 371 | 1.000 | \mathbf{v}_{k}^{T} A \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO3390 | 371 | 1.000 | \mathbf{v}_{0} | | |
| johnston-linear-matrix-algebra_FO3391 | 371 | 1.000 | \mathbf{v}_{0}=\mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO3394 | 375 | 1.000 | \mathbf{v}_{k}=A \mathbf{v}_{k-1} /\left\|A \mathbf{v}_{k-1}\right\| | | |
| johnston-linear-matrix-algebra_FO3396 | 372 | 0.997 | \mathbf{v}_{6}, \mathbf{v}_{7}, \mathbf{v}_{8} | | |
| johnston-linear-matrix-algebra_FO3397 | 372 | 1.000 | \mathbf{v} \approx \mathbf{v}_{5} \approx(0.37,0.93) | | |
| johnston-linear-matrix-algebra_FO3398 | 372 | 0.942 | \mathbf{v}=(2,5) / \sqrt{29} \approx(0.37,0.93) | | |
| johnston-linear-matrix-algebra_FO3399 | 372 | 1.000 | \mathbf{v}_{0}, \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO3400 | 373 | 1.000 | \mathbf{v}_{7} \approx(-0.26,-0.41,0.49,0.38,0.49,-0.38,-0.10) | | |
| johnston-linear-matrix-algebra_FO3401 | 373 | 1.000 | \lambda=-4.92 | | |
| johnston-linear-matrix-algebra_FO3402 | 373 | 0.999 | s-1 | | |
| johnston-linear-matrix-algebra_FO3403 | 374 | 1.000 | p_{C}(\lambda)=(-1)^{n} p(\lambda) | | |
| johnston-linear-matrix-algebra_FO3404 | 374 | 1.000 | C \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO3405 | 374 | 0.976 | 2 \times 2,3 \times 3 | | |
| johnston-linear-matrix-algebra_FO3406 | 374 | 1.000 | p(x)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{1} \lambda+a_{0} | | |
| johnston-linear-matrix-algebra_FO3407 | 374 | 0.999 | 2 n-1 | | |
| johnston-linear-matrix-algebra_FO3408 | 374 | 1.000 | p(x)=x^{6}-2 x^{5}+2 x^{4}-3 x^{3}+x^{2}+x-2 | | |
| johnston-linear-matrix-algebra_FO3411 | 374 | 0.992 | 1.68,-0.72,-0.17 \pm 1.32 i | | |
| johnston-linear-matrix-algebra_FO3412 | 374 | 1.000 | 0.69 \pm 0.67 i | | |
| johnston-linear-matrix-algebra_FO3413 | 375 | 1.000 | (1,1) / \sqrt{2} | | |
| johnston-linear-matrix-algebra_FO3414 | 375 | 1.000 | (1,3) / \sqrt{10} | | |
| johnston-linear-matrix-algebra_FO3415 | 375 | 1.000 | k=1,2,3, \ldots | | |
| johnston-linear-matrix-algebra_FO3416 | 375 | 1.000 | \mathbf{v}_{0}=(0.80,0.60) | | |
| johnston-linear-matrix-algebra_FO3417 | 375 | 1.000 | A=\left[\begin{array}{cc}1 & 1 \\ 3 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3418 | 376 | 1.000 | y=-3 x | | |
| johnston-linear-matrix-algebra_FO3419 | 376 | 1.000 | A^{2}=4 I | | |
| johnston-linear-matrix-algebra_FO3420 | 376 | 1.000 | \mathbf{v}_{1}=A \mathbf{v}_{0} /\left\|A \mathbf{v}_{0}\right\| | | |
| johnston-linear-matrix-algebra_FO3421 | 376 | 1.000 | \mathbf{v}_{2}, \mathbf{v}_{3} | | |
| johnston-linear-matrix-algebra_FO3422 | 376 | 1.000 | |\lambda|=|\bar{\lambda}| | | |
| johnston-linear-matrix-algebra_FO3423 | 376 | 0.999 | 1 \pm i | | |
| johnston-linear-matrix-algebra_FO3424 | 377 | 1.000 | \mathbf{v}_{0} \in \mathbb{C}_{n} | | |
| johnston-linear-matrix-algebra_FO3425 | 377 | 1.000 | |1+i|=|1-i|=\sqrt{2} | | |
| johnston-linear-matrix-algebra_FO3426 | 377 | 1.000 | \left\{\mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{n}\right\} | | |
| johnston-linear-matrix-algebra_FO3427 | 377 | 1.000 | \left|\lambda_{1}\right|>\left|\lambda_{j}\right| | | |
| johnston-linear-matrix-algebra_FO3428 | 377 | 1.000 | 2 \leq j \leq n | | |
| johnston-linear-matrix-algebra_FO3429 | 377 | 1.000 | \mathbf{v}_{0} \notin \operatorname{span}\left\{\mathbf{w}_{2}, \mathbf{w}_{3}, \ldots, \mathbf{w}_{n}\right\} | | |
| johnston-linear-matrix-algebra_FO3430 | 377 | 1.000 | \lim _{k \rightarrow \infty} \mathbf{v}_{k}^{*} A \mathbf{v}_{k}=\lambda_{1} | | |
| johnston-linear-matrix-algebra_FO3431 | 377 | 1.000 | \mathbf{v}_{k}=A^{k} \mathbf{v}_{0} /\left\|A^{k} \mathbf{v}_{0}\right\| | | |
| johnston-linear-matrix-algebra_FO3432 | 377 | 1.000 | A^{k} \mathbf{v}_{0} | | |
| johnston-linear-matrix-algebra_FO3433 | 377 | 1.000 | \mathbf{w}_{j} | | |
| johnston-linear-matrix-algebra_FO3434 | 377 | 1.000 | A \mathbf{w}_{j}=\lambda_{j} \mathbf{w}_{j} | | |
| johnston-linear-matrix-algebra_FO3435 | 377 | 0.985 | \left|\lambda_{j} / \lambda_{1}\right|<1 | | |
| johnston-linear-matrix-algebra_FO3436 | 377 | 1.000 | \left(\lambda_{j} / \lambda_{1}\right)^{k} \rightarrow 0 | | |
| johnston-linear-matrix-algebra_FO3437 | 377 | 1.000 | k \rightarrow \infty | | |
| johnston-linear-matrix-algebra_FO3438 | 377 | 1.000 | \mathbf{r}_{k} | | |
| johnston-linear-matrix-algebra_FO3439 | 378 | 1.000 | \left|\lambda_{1}\right|^{2 k} | | |
| johnston-linear-matrix-algebra_FO3440 | 378 | 0.994 | \bar{\lambda}_{1}^{k} | | |
| johnston-linear-matrix-algebra_FO3441 | 378 | 1.000 | \mathbf{v}_{k}^{*} | | |
| johnston-linear-matrix-algebra_FO3442 | 378 | 1.000 | \lambda_{1}^{k} | | |
| johnston-linear-matrix-algebra_FO3443 | 378 | 1.000 | \lim _{k \rightarrow \infty} \mathbf{r}_{k}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3444 | 378 | 1.000 | \mathbf{w}_{1} | | |
| johnston-linear-matrix-algebra_FO3445 | 378 | 1.000 | \left(\lambda_{1} /\left|\lambda_{1}\right|\right)^{k} | | |
| johnston-linear-matrix-algebra_FO3446 | 378 | 0.990 | \lambda_{1} /\left|\lambda_{1}\right|=-1 | | |
| johnston-linear-matrix-algebra_FO3447 | 378 | 1.000 | \left|\lambda_{2} / \lambda_{1}\right| | | |
| johnston-linear-matrix-algebra_FO3448 | 378 | 1.000 | A=\left[\begin{array}{ccc}8 & -1 & 0 \\ -1 & 2 & 5 \\ 0 & 5 & -6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3449 | 379 | 1.000 | \mathbf{v}_{318} | | |
| johnston-linear-matrix-algebra_FO3450 | 379 | 1.000 | \mathbf{v}_{320} | | |
| johnston-linear-matrix-algebra_FO3451 | 379 | 1.000 | \mathbf{v}_{319} | | |
| johnston-linear-matrix-algebra_FO3452 | 379 | 1.000 | \mathbf{v}_{321} | | |
| johnston-linear-matrix-algebra_FO3453 | 379 | 0.990 | \lambda_{3} \approx 4.19 | | |
| johnston-linear-matrix-algebra_FO3454 | 379 | 1.000 | A-\lambda_{1} \mathbf{w}_{1} \mathbf{w}_{1}^{T} | | |
| johnston-linear-matrix-algebra_FO3455 | 379 | 1.000 | k=63 | | |
| johnston-linear-matrix-algebra_FO3456 | 379 | 0.803 | k=156 | | |
| johnston-linear-matrix-algebra_FO3457 | 379 | 1.000 | k=320 | | |
| johnston-linear-matrix-algebra_FO3458 | 379 | 0.999 | \lambda_{1} \approx-8.41 | | |
| johnston-linear-matrix-algebra_FO3459 | 379 | 0.999 | \lambda_{2} \approx 8.22 | | |
| johnston-linear-matrix-algebra_FO3460 | 379 | 0.942 | \left|\lambda_{2} / \lambda_{1}\right| \approx 0.98 | | |
| johnston-linear-matrix-algebra_FO3461 | 379 | 0.763 | \operatorname{span}\{(0.97,-0.22,-0.08)\} | | |
| johnston-linear-matrix-algebra_FO3462 | 379 | 1.000 | \mathbf{v}_{k}^{*} A \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO3463 | 380 | 1.000 | \mathbf{w}_{2} | | |
| johnston-linear-matrix-algebra_FO3464 | 380 | 0.993 | A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 4 \\ 3 & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3465 | 380 | 1.000 | \lambda_{1} \approx 8.53 | | |
| johnston-linear-matrix-algebra_FO3466 | 380 | 1.000 | \mathbf{w}_{1} \approx(0.38,0.85,0.37) | | |
| johnston-linear-matrix-algebra_FO3469 | 380 | 1.000 | \lambda_{2} \approx-2.00 | | |
| johnston-linear-matrix-algebra_FO3470 | 380 | 1.000 | \mathbf{w}_{2} \approx(0.72,0.03,-0.69) | | |
| johnston-linear-matrix-algebra_FO3471 | 381 | 1.000 | \lambda_{3} \approx 0.47 | | |
| johnston-linear-matrix-algebra_FO3472 | 381 | 1.000 | -2,(9+\sqrt{65}) / 2 \approx 8.53 | | |
| johnston-linear-matrix-algebra_FO3473 | 381 | 1.000 | (9-\sqrt{65}) / 2 \approx 0.47 | | |
| johnston-linear-matrix-algebra_FO3474 | 381 | 1.000 | 6.25,-2.55 \pm 1.88 i | | |
| johnston-linear-matrix-algebra_FO3475 | 381 | 1.000 | -2.55 \pm 1.88 i | | |
| johnston-linear-matrix-algebra_FO3480 | 382 | 1.000 | [A \mathbf{v}]_{j} | | |
| johnston-linear-matrix-algebra_FO3481 | 382 | 0.662 | \lambda_{1}>\left|\lambda_{j}\right| | | |
| johnston-linear-matrix-algebra_FO3482 | 382 | 1.000 | \lambda_{j}(2 \leq j \leq n) | | |
| johnston-linear-matrix-algebra_FO3483 | 382 | 1.000 | \mathbf{v} \geq \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO3484 | 382 | 1.000 | \mathbf{v}>\mathbf{w} | | |
| johnston-linear-matrix-algebra_FO3485 | 382 | 1.000 | v_{j} \geq w_{j} | | |
| johnston-linear-matrix-algebra_FO3486 | 382 | 0.999 | v_{j}>w_{j} | | |
| johnston-linear-matrix-algebra_FO3487 | 382 | 0.874 | \mathbf{v} \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3488 | 382 | 1.000 | \mathbf{v}>\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3489 | 382 | 1.000 | S^{n} \subset \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO3490 | 382 | 1.000 | S^{n} | | |
| johnston-linear-matrix-algebra_FO3491 | 382 | 1.000 | \mathbf{v} \in S^{n} | | |
| johnston-linear-matrix-algebra_FO3492 | 382 | 1.000 | A \mathbf{v}>\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3493 | 382 | 1.000 | L: S^{n} \rightarrow \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO3494 | 382 | 1.000 | c>0 | | |
| johnston-linear-matrix-algebra_FO3495 | 382 | 1.000 | A \mathbf{v} \geq c \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3496 | 382 | 1.000 | L(\mathbf{v})>0 | | |
| johnston-linear-matrix-algebra_FO3497 | 382 | 1.000 | L(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO3498 | 383 | 1.000 | \left|[B \mathbf{w}]_{i}\right| | | |
| johnston-linear-matrix-algebra_FO3499 | 383 | 1.000 | B \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO3500 | 383 | 0.998 | [B|\mathbf{w}|]_{i} | | |
| johnston-linear-matrix-algebra_FO3501 | 383 | 1.000 | B|\mathbf{w}| | | |
| johnston-linear-matrix-algebra_FO3502 | 383 | 1.000 | \mu \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO3503 | 383 | 1.000 | \mathbf{v}_{*} \in S^{n} | | |
| johnston-linear-matrix-algebra_FO3504 | 383 | 1.000 | L\left(\mathbf{v}_{*}\right)=\mu \geq L(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO3505 | 383 | 0.998 | A, \mathbf{v}_{*} | | |
| johnston-linear-matrix-algebra_FO3506 | 383 | 0.998 | \mu | | |
| johnston-linear-matrix-algebra_FO3507 | 383 | 1.000 | L(\mathbf{v})=\mu | | |
| johnston-linear-matrix-algebra_FO3508 | 383 | 1.000 | A \mathbf{v}=\mu \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3509 | 383 | 1.000 | A \mathbf{v} \geq \mu \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3510 | 383 | 1.000 | A \mathbf{v}-\mu \mathbf{v} \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3511 | 383 | 0.985 | A \mathbf{v}-\mu \mathbf{v} \neq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3512 | 383 | 1.000 | A(A \mathbf{v}-\mu \mathbf{v})>\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3513 | 383 | 1.000 | A(A \mathbf{v})>\mu A \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3514 | 383 | 1.000 | \mu \geq L(A \mathbf{v} /\|A \mathbf{v}\|) | | |
| johnston-linear-matrix-algebra_FO3515 | 383 | 1.000 | \mathbf{v}^{*} | | |
| johnston-linear-matrix-algebra_FO3516 | 383 | 1.000 | 1 \leq i \leq n | | |
| johnston-linear-matrix-algebra_FO3517 | 383 | 1.000 | [A \mathbf{v}]_{i}=a_{i, 1} v_{2}+a_{i, 2} v_{2}+\cdots+a_{i, n} v_{n} | | |
| johnston-linear-matrix-algebra_FO3518 | 383 | 1.000 | \mathbf{v}=(A \mathbf{v}) / \mu>\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3519 | 383 | 1.000 | \mathbf{v}_{*} | | |
| johnston-linear-matrix-algebra_FO3520 | 383 | 1.000 | \mu=\lambda_{1} | | |
| johnston-linear-matrix-algebra_FO3521 | 383 | 1.000 | B \in \mathcal{M}_{n}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3522 | 383 | 1.000 | O \leq B \leq A | | |
| johnston-linear-matrix-algebra_FO3523 | 383 | 1.000 | |\lambda| \leq \mu | | |
| johnston-linear-matrix-algebra_FO3524 | 383 | 0.997 | |\mathbf{w}| | | |
| johnston-linear-matrix-algebra_FO3525 | 383 | 0.996 | |\mathbf{w}|=\left(\left|w_{1}\right|,\left|w_{2}\right|, \ldots,\left|w_{n}\right|\right) | | |
| johnston-linear-matrix-algebra_FO3526 | 383 | 1.000 | B|\mathbf{w}| \geq|\lambda||\mathbf{w}| | | |
| johnston-linear-matrix-algebra_FO3527 | 383 | 1.000 | (A-B)|\mathbf{w}| \geq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3528 | 383 | 1.000 | A|\mathbf{w}| \geq B|\mathbf{w}| \geq|\lambda||\mathbf{w}| | | |
| johnston-linear-matrix-algebra_FO3529 | 383 | 1.000 | L(|\mathbf{w}|) \geq|\lambda| | | |
| johnston-linear-matrix-algebra_FO3530 | 383 | 1.000 | \mu \geq L(|\mathbf{w}|) | | |
| johnston-linear-matrix-algebra_FO3531 | 383 | 1.000 | \mu \geq|\lambda| | | |
| johnston-linear-matrix-algebra_FO3532 | 384 | 1.000 | \mu \leq \lambda | | |
| johnston-linear-matrix-algebra_FO3533 | 384 | 1.000 | \mu \geq \lambda | | |
| johnston-linear-matrix-algebra_FO3534 | 384 | 1.000 | \mathbf{v}_{1}>c \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3535 | 384 | 1.000 | B=A | | |
| johnston-linear-matrix-algebra_FO3536 | 384 | 1.000 | \lambda=\mu | | |
| johnston-linear-matrix-algebra_FO3537 | 384 | 1.000 | \lambda>0 | | |
| johnston-linear-matrix-algebra_FO3538 | 384 | 1.000 | c \mu \geq c \lambda | | |
| johnston-linear-matrix-algebra_FO3539 | 384 | 1.000 | A\left(\mathbf{v}_{1}-c \mathbf{v}\right)= | | |
| johnston-linear-matrix-algebra_FO3540 | 384 | 1.000 | \mu\left(\mathbf{v}_{1}-c \mathbf{v}\right) | | |
| johnston-linear-matrix-algebra_FO3541 | 384 | 1.000 | \mathbf{v}_{1}-c \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3542 | 384 | 1.000 | |\lambda|=\mu | | |
| johnston-linear-matrix-algebra_FO3543 | 384 | 1.000 | b_{i, j} w_{j} | | |
| johnston-linear-matrix-algebra_FO3544 | 384 | 1.000 | b_{i, j}>0 | | |
| johnston-linear-matrix-algebra_FO3545 | 385 | 1.000 | \mathbf{v}_{5}=(0.58,0.58,0.58) | | |
| johnston-linear-matrix-algebra_FO3546 | 385 | 1.000 | \lambda=7 | | |
| johnston-linear-matrix-algebra_FO3547 | 385 | 1.000 | \lambda \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3548 | 385 | 1.000 | n \approx 5 | | |
| johnston-linear-matrix-algebra_FO3549 | 385 | 1.000 | r_{i} | | |
| johnston-linear-matrix-algebra_FO3550 | 385 | 1.000 | i, p_{j} | | |
| johnston-linear-matrix-algebra_FO3551 | 386 | 0.915 | \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D} | | |
| johnston-linear-matrix-algebra_FO3552 | 386 | 1.000 | r_{A}, r_{B}, r_{C}, r_{D} | | |
| johnston-linear-matrix-algebra_FO3553 | 386 | 1.000 | r_{E} | | |
| johnston-linear-matrix-algebra_FO3554 | 386 | 0.999 | \left(r_{A}, r_{B}, r_{C}, r_{D}, r_{E}\right)= | | |
| johnston-linear-matrix-algebra_FO3555 | 386 | 1.000 | r_{A}>r_{B}=r_{C}>r_{E}>r_{D} | | |
| johnston-linear-matrix-algebra_FO3556 | 386 | 1.000 | 1 / p_{j} | | |
| johnston-linear-matrix-algebra_FO3557 | 386 | 1.000 | p_{j} | | |
| johnston-linear-matrix-algebra_FO3558 | 387 | 1.000 | A \mathbf{r}=\mathbf{r} | | |
| johnston-linear-matrix-algebra_FO3559 | 387 | 1.000 | \mathbf{r}=\left(r_{A}, r_{B}, r_{C}, r_{D}, r_{E}\right) | | |
| johnston-linear-matrix-algebra_FO3560 | 388 | 1.000 | \lambda_{2}=-0.77 | | |
| johnston-linear-matrix-algebra_FO3561 | 388 | 0.562 | \lambda_{1}=1 \mathrm{in} | | |
| johnston-linear-matrix-algebra_FO3562 | 388 | 0.998 | (-1 \pm i \sqrt{3}) / 2 | | |
| johnston-linear-matrix-algebra_FO3563 | 388 | 1.000 | A+\varepsilon J | | |
| johnston-linear-matrix-algebra_FO3564 | 388 | 1.000 | J | | |
| johnston-linear-matrix-algebra_FO3565 | 388 | 1.000 | \varepsilon | | |
| johnston-linear-matrix-algebra_FO3566 | 389 | 1.000 | \left[\begin{array}{cc}3 & -1 \\ -4 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3567 | 389 | 0.877 | \left[\begin{array}{ccc}-2 & 1 & 1 \\ 4 & 1 & 2 \\ 1 & 0 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3568 | 389 | 1.000 | \left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 4 & 1 \\ -4 & 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3569 | 389 | 0.497 | \left[\begin{array}{llll}3 & 2 & 4 & 3 \\ 1 & 1 & 1 & 3 \\ 3 & 4 & 2 & 0 \\ 0 & 3 & 1 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3570 | 389 | 0.999 | \left[\begin{array}{llll}0 & 5 & 5 & 2 \\ 5 & 4 & 7 & 2 \\ 5 & 4 & 0 & 6 \\ 7 & 5 & 5 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3571 | 389 | 1.000 | \left[\begin{array}{cc}-2 & 4 \\ 5 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3572 | 389 | 1.000 | \left[\begin{array}{ccc}-2 & 3 & -1 \\ 5 & 2 & 3 \\ 3 & 3 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3573 | 389 | 0.990 | A \in \mathcal{M}_{3}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3574 | 389 | 1.000 | \mathbf{v}_{0}, \mathbf{v}_{1}, \mathbf{v}_{2} | | |
| johnston-linear-matrix-algebra_FO3575 | 389 | 1.000 | \mathbf{v}_{0}=\mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO3576 | 389 | 0.415 | * * 3 | | |
| johnston-linear-matrix-algebra_FO3577 | 389 | 1.000 | A \in \mathcal{M}_{2}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3578 | 390 | 0.495 | \left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3579 | 390 | 1.000 | \left[\begin{array}{cc}2 & 1 \\ 1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3580 | 390 | 0.534 | \left[\begin{array}{cc}i & i \\ i & i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3581 | 390 | 0.798 | \left[\begin{array}{ll}\sqrt{5}+i & \sqrt{5}-i \\ \sqrt{5}-i & \sqrt{5}+i\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3582 | 390 | 0.809 | *(\mathbf{e})\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3583 | 390 | 1.000 | \left[\begin{array}{ccc}0 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3584 | 390 | 1.000 | \left[A^{k}\right]_{i, j}>0 | | |
| johnston-linear-matrix-algebra_FO3585 | 390 | 0.972 | \left[\begin{array}{cc}-1 & 1 \\ 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3586 | 390 | 1.000 | \left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3587 | 390 | 0.817 | \left[\begin{array}{ll}0 & 1 \\ 2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3588 | 390 | 1.000 | \left[\begin{array}{cc}-2 & 1 \\ 1 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3589 | 390 | 1.000 | \left[\begin{array}{ccc}1 & -1 & 1 \\ -1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3590 | 390 | 0.997 | \left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3591 | 390 | 1.000 | A=\left[\begin{array}{cc}-3 & -2 \\ 4 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3592 | 391 | 1.000 | 4+(1-i)(-2-2 i)= | | |
| johnston-linear-matrix-algebra_FO3593 | 391 | 1.000 | 4+(-2+2 i-2 i-2)= | | |
| johnston-linear-matrix-algebra_FO3594 | 391 | 0.935 | 4-4=0 | | |
| johnston-linear-matrix-algebra_FO3595 | 391 | 1.000 | \bar{B}=B | | |
| johnston-linear-matrix-algebra_FO3596 | 391 | 1.000 | \lambda_{ \pm}=-1 \pm 2 i | | |
| johnston-linear-matrix-algebra_FO3597 | 391 | 1.000 | \mathbf{v}_{ \pm} | | |
| johnston-linear-matrix-algebra_FO3598 | 391 | 0.998 | (A-(-1 \pm 2 i) I) \mathbf{v}_{ \pm}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO3599 | 391 | 0.999 | \lambda_{+}=-1+2 i | | |
| johnston-linear-matrix-algebra_FO3600 | 391 | 1.000 | \mathbf{v}_{+}=(1,-1-i) | | |
| johnston-linear-matrix-algebra_FO3601 | 391 | 1.000 | \lambda_{-}=-1-2 i | | |
| johnston-linear-matrix-algebra_FO3602 | 391 | 1.000 | \mathbf{v}_{-}=(1,-1+i) | | |
| johnston-linear-matrix-algebra_FO3603 | 391 | 1.000 | \overline{-1+2 i}= | | |
| johnston-linear-matrix-algebra_FO3604 | 391 | 1.000 | -1-2 i | | |
| johnston-linear-matrix-algebra_FO3605 | 391 | 1.000 | \overline{(1,-1-i)}=(1,-1+i) | | |
| johnston-linear-matrix-algebra_FO3606 | 391 | 1.000 | B \mathbf{v}=\lambda \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO3607 | 391 | 1.000 | B \overline{\mathbf{v}} | | |
| johnston-linear-matrix-algebra_FO3608 | 391 | 1.000 | B \overline{\mathbf{v}}=\bar{\lambda} \overline{\mathbf{v}} | | |
| johnston-linear-matrix-algebra_FO3610 | 391 | 1.000 | \lambda=r e^{i \theta} | | |
| johnston-linear-matrix-algebra_FO3611 | 391 | 1.000 | r=\sqrt{a^{2}+b^{2}} | | |
| johnston-linear-matrix-algebra_FO3612 | 391 | 1.000 | \theta=\arctan (b / a) | | |
| johnston-linear-matrix-algebra_FO3613 | 391 | 1.000 | a>0 | | |
| johnston-linear-matrix-algebra_FO3614 | 391 | 1.000 | a \leq 0 | | |
| johnston-linear-matrix-algebra_FO3615 | 391 | 1.000 | e^{i \theta}=\cos (\theta)+i \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO3616 | 391 | 1.000 | \operatorname{Re}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO3617 | 391 | 1.000 | \operatorname{Im}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO3618 | 391 | 1.000 | \mathbf{v}=\mathbf{x}+i \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO3619 | 391 | 1.000 | \operatorname{Re}(\mathbf{v})=\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO3620 | 391 | 1.000 | \operatorname{Im}(\mathbf{v})=\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO3621 | 391 | 1.000 | \overline{\mathbf{v}}= | | |
| johnston-linear-matrix-algebra_FO3622 | 391 | 1.000 | \operatorname{Re}(\mathbf{v})-i \operatorname{Im}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO3628 | 392 | 1.000 | \pi | | |
| johnston-linear-matrix-algebra_FO3629 | 392 | 0.999 | r e^{i \theta} \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO3630 | 392 | 1.000 | \mathbf{v} \in \mathbb{C}^{2} | | |
| johnston-linear-matrix-algebra_FO3631 | 392 | 1.000 | Q=[\operatorname{Re}(\mathbf{v}) \mid-\operatorname{Im}(\mathbf{v})] | | |
| johnston-linear-matrix-algebra_FO3632 | 392 | 0.999 | B=\{\operatorname{Re}(\mathbf{v}),-\operatorname{Im}(\mathbf{v})\} | | |
| johnston-linear-matrix-algebra_FO3633 | 392 | 1.000 | \lambda=-1+2 i= | | |
| johnston-linear-matrix-algebra_FO3634 | 392 | 1.000 | \sqrt{5} e^{i \theta} | | |
| johnston-linear-matrix-algebra_FO3635 | 392 | 1.000 | \theta \approx \pm(0.6476) \pi | | |
| johnston-linear-matrix-algebra_FO3636 | 392 | 1.000 | \mathbf{v}=(1,-1-i) | | |
| johnston-linear-matrix-algebra_FO3637 | 393 | 1.000 | r=\sqrt{5} | | |
| johnston-linear-matrix-algebra_FO3638 | 393 | 1.000 | B=\{\operatorname{Re}(\mathbf{v}),-\operatorname{Im}(\mathbf{v})\}=\{(1,-1),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO3639 | 393 | 1.000 | \theta \approx | | |
| johnston-linear-matrix-algebra_FO3640 | 393 | 0.936 | (0.6476) \pi) | | |
| johnston-linear-matrix-algebra_FO3641 | 393 | 0.936 | r=\sqrt{5} \approx 2.2361 | | |
| johnston-linear-matrix-algebra_FO3643 | 393 | 1.000 | r e^{i \theta} | | |
| johnston-linear-matrix-algebra_FO3644 | 393 | 1.000 | r e^{-i \theta} | | |
| johnston-linear-matrix-algebra_FO3645 | 393 | 1.000 | H \in \mathcal{M}_{2}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO3646 | 394 | 0.913 | Q H=P | | |
| johnston-linear-matrix-algebra_FO3647 | 394 | 0.947 | H P^{-1}=Q^{-1} | | |
| johnston-linear-matrix-algebra_FO3648 | 394 | 0.927 | \sin (-\theta)=-\sin (\theta) | | |
| johnston-linear-matrix-algebra_FO3649 | 394 | 1.000 | e^{-i \theta}=\cos (\theta)-i \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO3650 | 394 | 1.000 | Q=P H^{-1} | | |
| johnston-linear-matrix-algebra_FO3651 | 394 | 1.000 | D=P^{-1} A P | | |
| johnston-linear-matrix-algebra_FO3652 | 394 | 1.000 | Q^{-1} A Q | | |
| johnston-linear-matrix-algebra_FO3653 | 394 | 1.000 | H D H^{-1} | | |
| johnston-linear-matrix-algebra_FO3654 | 394 | 1.000 | H D H^{-1}=r\left[R^{\theta}\right] | | |
| johnston-linear-matrix-algebra_FO3655 | 394 | 1.000 | H D=r\left[R^{\theta}\right] H | | |
| johnston-linear-matrix-algebra_FO3656 | 394 | 1.000 | H | | |
| johnston-linear-matrix-algebra_FO3657 | 394 | 1.000 | \lambda_{ \pm}=-1 \pm | | |
| johnston-linear-matrix-algebra_FO3658 | 394 | 1.000 | 2 i | | |
| johnston-linear-matrix-algebra_FO3659 | 394 | 1.000 | \mathbf{v}_{ \pm}=(1,-1 \mp i) | | |
| johnston-linear-matrix-algebra_FO3660 | 395 | 1.000 | \left[\operatorname{Re}\left(\mathbf{v}_{+}\right) \mid-\operatorname{Im}\left(\mathbf{v}_{+}\right)\right] | | |
| johnston-linear-matrix-algebra_FO3661 | 395 | 1.000 | \sqrt{5}\left[R^{\theta}\right] | | |
| johnston-linear-matrix-algebra_FO3662 | 395 | 1.000 | \theta=\arccos (-1 / \sqrt{5}) \approx(0.6476) \pi | | |
| johnston-linear-matrix-algebra_FO3663 | 395 | 1.000 | \operatorname{diag}(a, b, c, \ldots) | | |
| johnston-linear-matrix-algebra_FO3664 | 395 | 0.999 | a, b, c, \ldots | | |
| johnston-linear-matrix-algebra_FO3665 | 395 | 1.000 | A=\mathcal{M}_{n}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3666 | 395 | 1.000 | 2 \ell | | |
| johnston-linear-matrix-algebra_FO3667 | 395 | 0.951 | 2 \ell+m=n | | |
| johnston-linear-matrix-algebra_FO3668 | 395 | 1.000 | r_{j}, \lambda_{j}, \theta_{j} \in \mathbb{R}, \mathbf{w}_{j} \in \mathbb{R}^{n} | | |
| johnston-linear-matrix-algebra_FO3669 | 395 | 1.000 | \mathbf{v}_{j} \in \mathbb{C}^{n} | | |
| johnston-linear-matrix-algebra_FO3670 | 395 | 1.000 | A=Q B Q^{-1} | | |
| johnston-linear-matrix-algebra_FO3671 | 395 | 1.000 | B, Q \in \mathcal{M}_{n}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3672 | 395 | 1.000 | P, D \in \mathcal{M}_{n}(\mathbb{C}) | | |
| johnston-linear-matrix-algebra_FO3673 | 396 | 1.000 | \tilde{H} | | |
| johnston-linear-matrix-algebra_FO3674 | 396 | 0.993 | (1 \pm i) / 2 | | |
| johnston-linear-matrix-algebra_FO3675 | 396 | 1.000 | 1 \leq j \leq \ell | | |
| johnston-linear-matrix-algebra_FO3676 | 396 | 1.000 | Q_{j} H=P_{j} | | |
| johnston-linear-matrix-algebra_FO3677 | 396 | 1.000 | H D_{j}= | | |
| johnston-linear-matrix-algebra_FO3678 | 396 | 1.000 | r_{j}\left[R^{\theta_{j}}\right] H | | |
| johnston-linear-matrix-algebra_FO3679 | 396 | 0.999 | \tilde{H} D \tilde{H}^{-1}=B | | |
| johnston-linear-matrix-algebra_FO3680 | 397 | 1.000 | -\operatorname{Im}(\mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO3681 | 397 | 1.000 | p_{A}(\lambda)=-\lambda^{3}+3 \lambda^{2}-(5 / 2) \lambda+1 | | |
| johnston-linear-matrix-algebra_FO3682 | 397 | 0.672 | (1+i) / 2,(1-i) / 2 | | |
| johnston-linear-matrix-algebra_FO3683 | 397 | 1.000 | \mathbf{v}=(1, i, 1), \overline{\mathbf{v}}=(1,-i, 1) | | |
| johnston-linear-matrix-algebra_FO3684 | 397 | 1.000 | \mathbf{w}=(0,1,2) | | |
| johnston-linear-matrix-algebra_FO3685 | 397 | 0.980 | (1+i) / 2 | | |
| johnston-linear-matrix-algebra_FO3686 | 397 | 0.980 | r= | | |
| johnston-linear-matrix-algebra_FO3687 | 397 | 1.000 | \sqrt{(1 / 2)^{2}+(1 / 2)^{2}}=1 / \sqrt{2} | | |
| johnston-linear-matrix-algebra_FO3688 | 397 | 1.000 | \theta=\arctan ((1 / 2) /(1 / 2))=\arctan (1)= | | |
| johnston-linear-matrix-algebra_FO3689 | 397 | 1.000 | (1+i) / 2=\frac{1}{\sqrt{2}} e^{i \pi / 4} | | |
| johnston-linear-matrix-algebra_FO3690 | 397 | 0.539 | (1+ | | |
| johnston-linear-matrix-algebra_FO3691 | 397 | 1.000 | i) / 2 | | |
| johnston-linear-matrix-algebra_FO3692 | 397 | 1.000 | \mathbf{v}=(1, i, 1) | | |
| johnston-linear-matrix-algebra_FO3693 | 397 | 0.920 | (0,1,2), A | | |
| johnston-linear-matrix-algebra_FO3694 | 397 | 1.000 | \theta=\pi / 4 | | |
| johnston-linear-matrix-algebra_FO3695 | 398 | 1.000 | r=1 / \sqrt{2} | | |
| johnston-linear-matrix-algebra_FO3696 | 398 | 0.996 | \left[R^{\theta}\right]^{k}=\left[R^{k \theta}\right] | | |
| johnston-linear-matrix-algebra_FO3697 | 398 | 0.996 | k \theta | | |
| johnston-linear-matrix-algebra_FO3698 | 398 | 1.000 | A^{40} | | |
| johnston-linear-matrix-algebra_FO3699 | 398 | 1.000 | ((1+i) / 2)^{40} | | |
| johnston-linear-matrix-algebra_FO3700 | 398 | 1.000 | \left.((1+i) / 2)^{40}=1 / 2^{20}\right) | | |
| johnston-linear-matrix-algebra_FO3701 | 399 | 1.000 | \left[R^{40 \pi / 4}\right] | | |
| johnston-linear-matrix-algebra_FO3702 | 399 | 1.000 | \theta=40 \pi / 4=10 \pi | | |
| johnston-linear-matrix-algebra_FO3703 | 399 | 0.996 | r, \theta \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO3704 | 399 | 0.996 | Q \in \mathcal{M}_{2}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3705 | 399 | 1.000 | Q\left(r\left[R^{\theta}\right]\right) Q^{-1} | | |
| johnston-linear-matrix-algebra_FO3706 | 399 | 0.725 | *(\mathbf{a})\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3707 | 399 | 0.725 | *(\mathbf{c})\left[\begin{array}{cc}2 & 1 \\ -2 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3708 | 399 | 1.000 | \left[\begin{array}{cc}0 & 1 \\ -2 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3709 | 399 | 1.000 | \left[\begin{array}{cc}1 & -\sqrt{3} \\ \sqrt{3} & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3710 | 399 | 0.970 | \left[\begin{array}{ccc}1 & 0 & 0 \\ -1 & 0 & 1 \\ 1 & -1 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3711 | 399 | 1.000 | \left[\begin{array}{ccc}1 & -\sqrt{3} & \sqrt{3} \\ 0 & 3 & 0 \\ -\sqrt{3} & 2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3712 | 399 | 0.943 | \left[\begin{array}{cccc}1 & -2 & 2 & 1 \\ 0 & 1 & 0 & 0 \\ -1 & -1 & 2 & 1 \\ 1 & 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3713 | 399 | 1.000 | \left[\begin{array}{cccc}4 & -3 & -1 & 2 \\ 6 & -5 & 0 & 1 \\ 4 & -5 & 1 & 0 \\ 0 & -1 & 3 & -2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO3714 | 399 | 1.000 | A \in \mathcal{M}_{4}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3715 | 399 | 1.000 | \left[R^{\theta}\right] \in \mathcal{M}_{2}(\mathbb{R}) | | |
| johnston-linear-matrix-algebra_FO3716 | 400 | 1.000 | x_{0} | | |
| johnston-linear-matrix-algebra_FO3717 | 400 | 1.000 | x_{1}, x_{2}, x_{3}, \ldots | | |
| johnston-linear-matrix-algebra_FO3718 | 400 | 1.000 | x_{n} | | |
| johnston-linear-matrix-algebra_FO3719 | 400 | 1.000 | a_{0}, a_{1}, \ldots, a_{k-1} | | |
| johnston-linear-matrix-algebra_FO3720 | 400 | 0.994 | F_{0}, F_{1}, F_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO3721 | 400 | 0.907 | F_{n}=F_{n-1}+F_{n-2} | | |
| johnston-linear-matrix-algebra_FO3722 | 400 | 1.000 | F_{0}=0 | | |
| johnston-linear-matrix-algebra_FO3723 | 400 | 1.000 | F_{1}=1 | | |
| johnston-linear-matrix-algebra_FO3724 | 400 | 1.000 | x_{0}=3, x_{1}=-2 | | |
| johnston-linear-matrix-algebra_FO3725 | 400 | 1.000 | x_{2}=8 | | |
| johnston-linear-matrix-algebra_FO3726 | 400 | 1.000 | x_{6}, x_{7} | | |
| johnston-linear-matrix-algebra_FO3727 | 401 | 1.000 | x_{n}=x_{n-1}+x_{n-2} | | |
| johnston-linear-matrix-algebra_FO3728 | 401 | 1.000 | x_{n}=4 x_{n-1}-x_{n-2}-6 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3729 | 401 | 1.000 | x_{n}=x_{n-1}+4 x_{n-3}+2 x_{n-4} | | |
| johnston-linear-matrix-algebra_FO3730 | 401 | 1.000 | a_{1}=a_{0}=1 | | |
| johnston-linear-matrix-algebra_FO3731 | 401 | 0.877 | a_{2}=4, a_{1}=-1 | | |
| johnston-linear-matrix-algebra_FO3732 | 401 | 1.000 | a_{0}=-6 | | |
| johnston-linear-matrix-algebra_FO3733 | 401 | 0.948 | a_{3}=1, a_{2}=0, a_{1}=4 | | |
| johnston-linear-matrix-algebra_FO3734 | 401 | 0.948 | a_{0}=2 | | |
| johnston-linear-matrix-algebra_FO3735 | 401 | 0.999 | x_{0}, x_{1}, x_{2}, \ldots | | |
| johnston-linear-matrix-algebra_FO3736 | 401 | 1.000 | n=0,1,2, \ldots | | |
| johnston-linear-matrix-algebra_FO3737 | 401 | 1.000 | \mathbf{x}_{n}= | | |
| johnston-linear-matrix-algebra_FO3738 | 401 | 1.000 | \left(x_{n}, x_{n+1}, \ldots, x_{n+k-1}\right) \in \mathbb{R}^{k} | | |
| johnston-linear-matrix-algebra_FO3739 | 402 | 1.000 | \mathbf{x}_{0}=\left(x_{0}, x_{1}, \ldots, x_{k-1}\right) | | |
| johnston-linear-matrix-algebra_FO3740 | 402 | 1.000 | n=k | | |
| johnston-linear-matrix-algebra_FO3741 | 402 | 1.000 | C^{n} \mathbf{x}_{0} | | |
| johnston-linear-matrix-algebra_FO3742 | 402 | 1.000 | C^{n-k+1} \mathbf{x}_{0} | | |
| johnston-linear-matrix-algebra_FO3743 | 403 | 1.000 | 3^{n}-2^{n}+3(-1)^{n} | | |
| johnston-linear-matrix-algebra_FO3744 | 403 | 1.000 | x_{n}=3^{n}-2^{n}+3(-1)^{n} | | |
| johnston-linear-matrix-algebra_FO3745 | 403 | 1.000 | p(\lambda)=(-1)^{k}\left(\lambda^{k}-a_{k-1} \lambda^{k-1}-\cdots-a_{1} \lambda-a_{0}\right) | | |
| johnston-linear-matrix-algebra_FO3746 | 404 | 1.000 | x_{n-j} | | |
| johnston-linear-matrix-algebra_FO3747 | 404 | 1.000 | \lambda^{k-j} | | |
| johnston-linear-matrix-algebra_FO3748 | 404 | 1.000 | c_{0}, c_{1}, \ldots, c_{k-1} | | |
| johnston-linear-matrix-algebra_FO3749 | 404 | 1.000 | n=0 | | |
| johnston-linear-matrix-algebra_FO3750 | 404 | 1.000 | \lambda_{0}, \lambda_{1}, \ldots, \lambda_{k-1} | | |
| johnston-linear-matrix-algebra_FO3751 | 404 | 1.000 | p(\lambda)=\lambda^{k}-a_{k-1} \lambda^{k-1}-a_{k-2} \lambda^{k-2}-\cdots- | | |
| johnston-linear-matrix-algebra_FO3752 | 404 | 1.000 | a_{1} \lambda-a_{0} | | |
| johnston-linear-matrix-algebra_FO3753 | 404 | 1.000 | 0^{n} | | |
| johnston-linear-matrix-algebra_FO3754 | 404 | 1.000 | a_{0}=0 | | |
| johnston-linear-matrix-algebra_FO3755 | 404 | 1.000 | k-1 | | |
| johnston-linear-matrix-algebra_FO3756 | 405 | 1.000 | \lambda_{0}=1, \lambda_{1}=2 | | |
| johnston-linear-matrix-algebra_FO3757 | 405 | 1.000 | \lambda_{2}=5 | | |
| johnston-linear-matrix-algebra_FO3758 | 405 | 1.000 | x_{0}=1, x_{1}=4 | | |
| johnston-linear-matrix-algebra_FO3759 | 405 | 1.000 | x_{2}=22 | | |
| johnston-linear-matrix-algebra_FO3760 | 405 | 1.000 | x_{k} | | |
| johnston-linear-matrix-algebra_FO3761 | 405 | 1.000 | x_{k+1} | | |
| johnston-linear-matrix-algebra_FO3762 | 405 | 1.000 | x_{k+2} | | |
| johnston-linear-matrix-algebra_FO3763 | 405 | 1.000 | \left(c_{0}, c_{1}, c_{2}\right)=(1,-1,1) | | |
| johnston-linear-matrix-algebra_FO3764 | 405 | 1.000 | \mathbf{x}_{n}=C^{n} \mathbf{x}_{0} | | |
| johnston-linear-matrix-algebra_FO3765 | 406 | 1.000 | V \mathbf{c}=\mathbf{x}_{0} | | |
| johnston-linear-matrix-algebra_FO3766 | 406 | 0.996 | V^{-1} \mathbf{x}_{0}=\mathbf{c} | | |
| johnston-linear-matrix-algebra_FO3767 | 406 | 1.000 | \lambda_{1}=i | | |
| johnston-linear-matrix-algebra_FO3768 | 406 | 1.000 | \lambda_{2}=\bar{i}=-i | | |
| johnston-linear-matrix-algebra_FO3769 | 406 | 1.000 | c_{1}=1-3 i | | |
| johnston-linear-matrix-algebra_FO3770 | 406 | 1.000 | c_{2}=\overline{1-3 i}=1+3 i | | |
| johnston-linear-matrix-algebra_FO3771 | 406 | 0.988 | \mathbf{c}=\left(c_{0}, c_{1}, \ldots, c_{k-1}\right) | | |
| johnston-linear-matrix-algebra_FO3772 | 406 | 1.000 | V D^{n} \mathbf{c} | | |
| johnston-linear-matrix-algebra_FO3773 | 406 | 1.000 | x_{0}=3, x_{1}=8 | | |
| johnston-linear-matrix-algebra_FO3774 | 406 | 1.000 | x_{2}=2 | | |
| johnston-linear-matrix-algebra_FO3775 | 406 | 1.000 | \lambda_{0}=2, \lambda_{1}=i | | |
| johnston-linear-matrix-algebra_FO3776 | 406 | 1.000 | \lambda_{2}=-i | | |
| johnston-linear-matrix-algebra_FO3777 | 406 | 1.000 | \left(c_{0}, c_{1}, c_{2}\right)=(1,1-3 i, 1+3 i) | | |
| johnston-linear-matrix-algebra_FO3778 | 406 | 0.814 | x_{n}=c_{0} \lambda_{0}^{n}+c_{1} \lambda_{1}^{n}+c_{2} \lambda_{2}^{n}=2^{n}+(1-3 i) i^{n}+(1+3 i)(-i)^{n} \quad | | |
| johnston-linear-matrix-algebra_FO3779 | 406 | 0.814 | \quad n \geq 0 | | |
| johnston-linear-matrix-algebra_FO3780 | 407 | 0.999 | r_{0}=\ldots=r_{m-1}=1 | | |
| johnston-linear-matrix-algebra_FO3781 | 407 | 1.000 | \lambda_{1}=i=e^{i \pi / 2} | | |
| johnston-linear-matrix-algebra_FO3782 | 407 | 1.000 | \lambda_{2}=-i=e^{-i \pi / 2} | | |
| johnston-linear-matrix-algebra_FO3783 | 407 | 1.000 | e^{i \theta}+e^{-i \theta}=2 \cos (\theta) | | |
| johnston-linear-matrix-algebra_FO3784 | 407 | 1.000 | e^{i \theta}-e^{-i \theta}=2 i \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO3785 | 407 | 1.000 | \lambda_{0}, \lambda_{1}, \ldots, \lambda_{m-1} | | |
| johnston-linear-matrix-algebra_FO3786 | 407 | 1.000 | r_{0}, r_{1}, \ldots, r_{m-1} | | |
| johnston-linear-matrix-algebra_FO3787 | 407 | 1.000 | q_{0}, q_{1}, \ldots, q_{m-1} | | |
| johnston-linear-matrix-algebra_FO3788 | 407 | 1.000 | r_{0}-1, r_{1}-1, \ldots, r_{m-1}-1 | | |
| johnston-linear-matrix-algebra_FO3789 | 408 | 0.874 | x_{n-1} | | |
| johnston-linear-matrix-algebra_FO3790 | 408 | 1.000 | V_{j} | | |
| johnston-linear-matrix-algebra_FO3791 | 408 | 1.000 | k \times r_{j} | | |
| johnston-linear-matrix-algebra_FO3792 | 408 | 1.000 | k \times\left(r_{0}+\cdots+r_{m-1}\right)= | | |
| johnston-linear-matrix-algebra_FO3793 | 408 | 1.000 | r_{j}=1 | | |
| johnston-linear-matrix-algebra_FO3794 | 408 | 1.000 | k \times 1 | | |
| johnston-linear-matrix-algebra_FO3795 | 408 | 1.000 | x_{0}=6, x_{1}=0 | | |
| johnston-linear-matrix-algebra_FO3796 | 408 | 1.000 | x_{2}=3 | | |
| johnston-linear-matrix-algebra_FO3797 | 408 | 1.000 | \lambda_{0}=2 | | |
| johnston-linear-matrix-algebra_FO3798 | 408 | 1.000 | q_{0} | | |
| johnston-linear-matrix-algebra_FO3799 | 408 | 1.000 | q_{1} | | |
| johnston-linear-matrix-algebra_FO3800 | 408 | 0.979 | q_{0}(x)=c_{0} | | |
| johnston-linear-matrix-algebra_FO3801 | 408 | 0.979 | q_{1}(x)=c_{1}+c_{2} x | | |
| johnston-linear-matrix-algebra_FO3802 | 408 | 1.000 | c_{0}, c_{1}, c_{2} | | |
| johnston-linear-matrix-algebra_FO3803 | 408 | 1.000 | n=0,1 | | |
| johnston-linear-matrix-algebra_FO3804 | 408 | 1.000 | \left(c_{0}, c_{1}, c_{2}\right)=(1,5,-3) | | |
| johnston-linear-matrix-algebra_FO3805 | 408 | 1.000 | q_{0}, q_{1}, \ldots, q_{k-1} | | |
| johnston-linear-matrix-algebra_FO3806 | 408 | 1.000 | q_{j}(n)=c_{j, 0}+c_{j, 1} n+\cdots+c_{j, r_{j}-1} n^{r_{j}-1} | | |
| johnston-linear-matrix-algebra_FO3807 | 408 | 1.000 | 0 \leq j<m | | |
| johnston-linear-matrix-algebra_FO3808 | 408 | 1.000 | c_{j, \ell} | | |
| johnston-linear-matrix-algebra_FO3809 | 409 | 1.000 | 0^{0}=1 | | |
| johnston-linear-matrix-algebra_FO3810 | 409 | 1.000 | d_{0} 0^{0} \lambda_{j}^{0}=d_{0} | | |
| johnston-linear-matrix-algebra_FO3811 | 409 | 0.999 | d_{0} 0^{\ell} \lambda_{j}^{0}=0 | | |
| johnston-linear-matrix-algebra_FO3812 | 409 | 1.000 | \ell>0 | | |
| johnston-linear-matrix-algebra_FO3813 | 409 | 0.999 | d_{0}, d_{1}, \ldots, d_{k-1} | | |
| johnston-linear-matrix-algebra_FO3814 | 409 | 1.000 | r_{0}+r_{1}+\cdots+r_{m-1}=k | | |
| johnston-linear-matrix-algebra_FO3815 | 409 | 1.000 | d_{0}=d_{1}=\cdots=d_{k-1}=0 | | |
| johnston-linear-matrix-algebra_FO3816 | 409 | 1.000 | x_{0}=0, x_{1}=3, x_{2}=8 | | |
| johnston-linear-matrix-algebra_FO3817 | 409 | 1.000 | x_{3}=7 | | |
| johnston-linear-matrix-algebra_FO3818 | 410 | 1.000 | 1^{n}=1 | | |
| johnston-linear-matrix-algebra_FO3819 | 410 | 1.000 | \left(c_{0}+c_{1} n+c_{2} n^{2}\right) 1^{n} | | |
| johnston-linear-matrix-algebra_FO3820 | 410 | 1.000 | c_{0}+c_{1} n+c_{2} n^{2} | | |
| johnston-linear-matrix-algebra_FO3821 | 410 | 1.000 | 1-\phi \approx-0.6180 | | |
| johnston-linear-matrix-algebra_FO3822 | 410 | 1.000 | (1-\phi)^{n} \rightarrow 0 | | |
| johnston-linear-matrix-algebra_FO3823 | 410 | 1.000 | n \rightarrow \infty | | |
| johnston-linear-matrix-algebra_FO3824 | 410 | 0.996 | \lambda_{0}=1 | | |
| johnston-linear-matrix-algebra_FO3825 | 410 | 0.996 | \lambda_{1}=3 | | |
| johnston-linear-matrix-algebra_FO3826 | 410 | 1.000 | q_{0}(x)=c_{0}+c_{1} x+c_{2} x^{2} | | |
| johnston-linear-matrix-algebra_FO3827 | 410 | 1.000 | q_{1}(x)=c_{3} | | |
| johnston-linear-matrix-algebra_FO3828 | 410 | 1.000 | c_{0}, c_{1}, c_{2}, c_{3} | | |
| johnston-linear-matrix-algebra_FO3829 | 410 | 1.000 | n=0,1,2 | | |
| johnston-linear-matrix-algebra_FO3830 | 410 | 1.000 | \left(c_{0}, c_{1}, c_{2}, c_{3}\right)=(1,2,3,-1) | | |
| johnston-linear-matrix-algebra_FO3831 | 410 | 1.000 | F_{0}=0, F_{1}=1 | | |
| johnston-linear-matrix-algebra_FO3833 | 410 | 1.000 | \phi^{n} | | |
| johnston-linear-matrix-algebra_FO3834 | 410 | 1.000 | (1-\phi)^{n} | | |
| johnston-linear-matrix-algebra_FO3835 | 410 | 1.000 | \phi^{n} / \sqrt{5} | | |
| johnston-linear-matrix-algebra_FO3836 | 411 | 1.000 | q_{0}(n) \lambda_{0}^{n} | | |
| johnston-linear-matrix-algebra_FO3837 | 411 | 1.000 | q | | |
| johnston-linear-matrix-algebra_FO3838 | 411 | 0.887 | q(n+1) | | |
| johnston-linear-matrix-algebra_FO3839 | 411 | 0.887 | q(n) | | |
| johnston-linear-matrix-algebra_FO3840 | 411 | 1.000 | x_{8}=6308 | | |
| johnston-linear-matrix-algebra_FO3841 | 411 | 1.000 | x_{7}=2056 | | |
| johnston-linear-matrix-algebra_FO3842 | 412 | 1.000 | x_{0}=0, x_{1}=0, x_{2}=1, x_{n}=x_{n-1}-4 x_{n-2}+4 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3843 | 412 | 1.000 | x_{0}=0, x_{1}=1, x_{2}=0, x_{n}=2 x_{n-1}+9 x_{n-2}-18 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3844 | 412 | 1.000 | 1,2 i | | |
| johnston-linear-matrix-algebra_FO3845 | 412 | 1.000 | -2 i | | |
| johnston-linear-matrix-algebra_FO3846 | 412 | 0.999 | x_{n} / x_{n-1}=1 | | |
| johnston-linear-matrix-algebra_FO3847 | 412 | 0.979 | x_{n} / x_{n-1} \approx-4 | | |
| johnston-linear-matrix-algebra_FO3848 | 412 | 1.000 | (2 i)^{2}=(-2 i)^{2}=-4 | | |
| johnston-linear-matrix-algebra_FO3849 | 412 | 1.000 | x_{n} / x_{n-1} | | |
| johnston-linear-matrix-algebra_FO3850 | 412 | 1.000 | 3^{2}=(-3)^{2}=9 | | |
| johnston-linear-matrix-algebra_FO3851 | 413 | 1.000 | x_{n-3}, x_{n-11}, x_{n-12} | | |
| johnston-linear-matrix-algebra_FO3852 | 413 | 0.915 | x_{n-45}, x_{n-54} | | |
| johnston-linear-matrix-algebra_FO3853 | 413 | 1.000 | x_{5}=6 | | |
| johnston-linear-matrix-algebra_FO3854 | 413 | 1.000 | p(\lambda)=(\lambda+1)(\lambda-1)^{2} q(\lambda) | | |
| johnston-linear-matrix-algebra_FO3855 | 413 | 1.000 | q(\lambda) | | |
| johnston-linear-matrix-algebra_FO3856 | 414 | 0.992 | x_{n}=x_{n-1}+6 x_{n-2} | | |
| johnston-linear-matrix-algebra_FO3857 | 414 | 1.000 | x_{n}=6 x_{n-1}-9 x_{n-2} | | |
| johnston-linear-matrix-algebra_FO3858 | 414 | 0.995 | x_{n}=6 x_{n-1}-4 x_{n-2} | | |
| johnston-linear-matrix-algebra_FO3859 | 414 | 1.000 | x_{n}=2 x_{n-1}-2 x_{n-2} | | |
| johnston-linear-matrix-algebra_FO3860 | 414 | 1.000 | x_{n}=6 x_{n-1}-11 x_{n-2}+6 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3861 | 414 | 1.000 | x_{n}=2 x_{n-2}+x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3862 | 414 | 0.982 | x_{n}=6 x_{n-1}-12 x_{n-2}+8 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3863 | 414 | 1.000 | x_{n}=7 x_{n-1}-8 x_{n-2}-16 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3864 | 414 | 1.000 | x_{n}=5 x_{n-1}-4 x_{n-2}-6 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3865 | 414 | 1.000 | x_{n}=7 x_{n-1}-17 x_{n-2}+15 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3866 | 414 | 0.999 | x_{0}=4, x_{1}=-3 | | |
| johnston-linear-matrix-algebra_FO3867 | 414 | 1.000 | x_{0}=1, x_{1}=0 | | |
| johnston-linear-matrix-algebra_FO3868 | 414 | 0.993 | x_{0}=2, x_{1}=0 | | |
| johnston-linear-matrix-algebra_FO3869 | 414 | 1.000 | x_{0}=2, x_{1}=3 | | |
| johnston-linear-matrix-algebra_FO3870 | 414 | 0.997 | x_{0}=0, x_{1}=2, x_{2}=8 | | |
| johnston-linear-matrix-algebra_FO3871 | 414 | 1.000 | x_{0}=0, x_{1}=2, x_{2}=16 | | |
| johnston-linear-matrix-algebra_FO3872 | 414 | 1.000 | x_{0}=-2, x_{1}=1, x_{2}=5 | | |
| johnston-linear-matrix-algebra_FO3873 | 415 | 1.000 | x_{0}=7, x_{1}=8, x_{2}=4 | | |
| johnston-linear-matrix-algebra_FO3874 | 415 | 1.000 | x_{n}=4 x_{n-1}-6 x_{n-2}+4 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3875 | 415 | 1.000 | x_{n}=2^{n}-3^{n} | | |
| johnston-linear-matrix-algebra_FO3876 | 415 | 0.976 | x_{n}=n^{2}-n^{3} | | |
| johnston-linear-matrix-algebra_FO3877 | 415 | 1.000 | x_{n}=\sqrt{2 n}- | | |
| johnston-linear-matrix-algebra_FO3878 | 415 | 1.000 | \sqrt{3 n} | | |
| johnston-linear-matrix-algebra_FO3879 | 415 | 0.916 | \lim _{n \rightarrow \infty} x_{n+1} / x_{n}=4 | | |
| johnston-linear-matrix-algebra_FO3880 | 415 | 0.853 | x_{n}=2^{n}+3^{n} | | |
| johnston-linear-matrix-algebra_FO3881 | 415 | 1.000 | x_{n}=2^{n}-3^{n+1} | | |
| johnston-linear-matrix-algebra_FO3882 | 415 | 0.977 | x_{n}=2^{n}+3^{n}+4^{n} | | |
| johnston-linear-matrix-algebra_FO3883 | 415 | 1.000 | x_{n}=(1+n) 2^{n}+3^{n} | | |
| johnston-linear-matrix-algebra_FO3884 | 415 | 0.999 | x_{n}=\left(1+n+n^{2}\right) 2^{n}+3^{n} | | |
| johnston-linear-matrix-algebra_FO3885 | 415 | 1.000 | x_{n}=7+2^{n} | | |
| johnston-linear-matrix-algebra_FO3886 | 415 | 0.975 | x_{n}=n^{2} | | |
| johnston-linear-matrix-algebra_FO3887 | 415 | 1.000 | x_{n}=2(1+i)^{n}+2(1-i)^{n} | | |
| johnston-linear-matrix-algebra_FO3888 | 415 | 0.808 | x_{n}=(n-5) 3^{n}+(3+i)(1-2 i)^{n}+(3-i)(1+2 i)^{n} | | |
| johnston-linear-matrix-algebra_FO3889 | 415 | 1.000 | x_{n}=2^{n} \cos (\pi n / 4)-2^{n} \sin (\pi n / 4) | | |
| johnston-linear-matrix-algebra_FO3890 | 415 | 1.000 | x_{n}=-x_{n-2} | | |
| johnston-linear-matrix-algebra_FO3891 | 415 | 1.000 | x_{0}=4, x_{1}=4 | | |
| johnston-linear-matrix-algebra_FO3892 | 415 | 1.000 | x_{0}=3, x_{1}=1, x_{2}=-7 | | |
| johnston-linear-matrix-algebra_FO3893 | 415 | 1.000 | x_{n}=x_{n-1}-4 x_{n-2}+4 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3894 | 415 | 1.000 | x_{0}=-1, x_{1}=0, x_{2}=8 | | |
| johnston-linear-matrix-algebra_FO3895 | 415 | 1.000 | x_{n}=4 x_{n-1}-8 x_{n-2}+8 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3896 | 415 | 1.000 | x_{n}=\left\lfloor(4+\sqrt{11})^{n}\right\rfloor | | |
| johnston-linear-matrix-algebra_FO3897 | 415 | 1.000 | \lfloor\cdot\rfloor | | |
| johnston-linear-matrix-algebra_FO3898 | 415 | 1.000 | 4+\sqrt{11} | | |
| johnston-linear-matrix-algebra_FO3899 | 415 | 1.000 | x_{n}=x_{n-1}- | | |
| johnston-linear-matrix-algebra_FO3900 | 415 | 1.000 | 4 x_{n-2}+4 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO3901 | 415 | 1.000 | x_{0}=0, x_{1}=0 | | |
| johnston-linear-matrix-algebra_FO3902 | 415 | 1.000 | x_{2}=1 | | |
| johnston-linear-matrix-algebra_FO3903 | 415 | 1.000 | x_{0}=x_{1}, x_{2}=x_{3} | | |
| johnston-linear-matrix-algebra_FO3904 | 417 | 1.000 | a b=b a, a+b=b+a | | |
| johnston-linear-matrix-algebra_FO3905 | 417 | 1.000 | a, b, c \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO3906 | 417 | 0.992 | \varepsilon \times 0=1 | | |
| johnston-linear-matrix-algebra_FO3907 | 417 | 0.992 | 1 / 0 | | |
| johnston-linear-matrix-algebra_FO3908 | 417 | 1.000 | \sqrt{-1} | | |
| johnston-linear-matrix-algebra_FO3909 | 417 | 1.000 | (a+b i)+(c+d i)=(a+c)+(b+d) i | | |
| johnston-linear-matrix-algebra_FO3910 | 417 | 1.000 | x^{2}-2 x+5=0 | | |
| johnston-linear-matrix-algebra_FO3911 | 417 | 1.000 | x=1+2 i | | |
| johnston-linear-matrix-algebra_FO3912 | 418 | 0.923 | a+b i \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO3913 | 418 | 0.923 | (a, b) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO3914 | 419 | 1.000 | (a+b i)(a-b i) | | |
| johnston-linear-matrix-algebra_FO3915 | 419 | 0.957 | (a b-a b) i=0 | | |
| johnston-linear-matrix-algebra_FO3916 | 419 | 1.000 | z=a+b i | | |
| johnston-linear-matrix-algebra_FO3917 | 419 | 1.000 | |z|^{2}=z \bar{z} | | |
| johnston-linear-matrix-algebra_FO3918 | 419 | 1.000 | z=|z| u | | |
| johnston-linear-matrix-algebra_FO3919 | 419 | 1.000 | |z| | | |
| johnston-linear-matrix-algebra_FO3920 | 419 | 1.000 | u | | |
| johnston-linear-matrix-algebra_FO3921 | 419 | 1.000 | \theta \in[0,2 \pi) | | |
| johnston-linear-matrix-algebra_FO3922 | 419 | 1.000 | 2 \pi | | |
| johnston-linear-matrix-algebra_FO3923 | 419 | 1.000 | \cos (\theta)+i \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO3924 | 419 | 1.000 | z= | | |
| johnston-linear-matrix-algebra_FO3925 | 419 | 1.000 | |z|(\cos (\theta)+i \sin (\theta)) | | |
| johnston-linear-matrix-algebra_FO3926 | 419 | 1.000 | \operatorname{Re}\left(e^{i \theta}\right)=\cos (\theta) | | |
| johnston-linear-matrix-algebra_FO3927 | 419 | 0.995 | \operatorname{Im}\left(e^{i \theta}\right)=\sin (\theta) | | |
| johnston-linear-matrix-algebra_FO3928 | 419 | 1.000 | e^{i \theta} | | |
| johnston-linear-matrix-algebra_FO3929 | 420 | 0.980 | |\cos (\theta)+i \sin (\theta)|^{2}= | | |
| johnston-linear-matrix-algebra_FO3930 | 420 | 1.000 | \cos ^{2}(\theta)+\sin ^{2}(\theta)=1 | | |
| johnston-linear-matrix-algebra_FO3931 | 420 | 0.964 | \operatorname{sign}(b)= \pm 1 | | |
| johnston-linear-matrix-algebra_FO3932 | 420 | 1.000 | b<0 | | |
| johnston-linear-matrix-algebra_FO3933 | 420 | 1.000 | -\pi<\theta<0 | | |
| johnston-linear-matrix-algebra_FO3934 | 420 | 1.000 | [0,2 \pi) | | |
| johnston-linear-matrix-algebra_FO3935 | 420 | 1.000 | e^{x}, \cos (x) | | |
| johnston-linear-matrix-algebra_FO3936 | 420 | 1.000 | x=i \theta | | |
| johnston-linear-matrix-algebra_FO3937 | 420 | 1.000 | z=r e^{i \theta} | | |
| johnston-linear-matrix-algebra_FO3938 | 420 | 1.000 | r=|z| | | |
| johnston-linear-matrix-algebra_FO3939 | 420 | 1.000 | \left(r_{1} e^{i \theta_{1}}\right)\left(r_{2} e^{i \theta_{2}}\right)= | | |
| johnston-linear-matrix-algebra_FO3940 | 420 | 1.000 | \left(r_{1} r_{2}\right) e^{i\left(\theta_{1}+\theta_{2}\right)} | | |
| johnston-linear-matrix-algebra_FO3941 | 420 | 1.000 | \left(r e^{i \theta}\right)^{n}=r^{n} e^{i n \theta} | | |
| johnston-linear-matrix-algebra_FO3942 | 421 | 1.000 | e^{2 \pi i}=e^{0 i}=1 | | |
| johnston-linear-matrix-algebra_FO3943 | 421 | 1.000 | z=i | | |
| johnston-linear-matrix-algebra_FO3944 | 421 | 1.000 | z=e^{\pi i / 2} | | |
| johnston-linear-matrix-algebra_FO3945 | 421 | 1.000 | e^{\pi i / 6}=(\sqrt{3}+i) / 2 | | |
| johnston-linear-matrix-algebra_FO3946 | 421 | 1.000 | r^{1 / n} e^{i \theta / n} | | |
| johnston-linear-matrix-algebra_FO3947 | 421 | 1.000 | z^{1 / n} | | |
| johnston-linear-matrix-algebra_FO3948 | 422 | 1.000 | p: \mathbb{C} \rightarrow \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO3949 | 422 | 1.000 | \theta=0 | | |
| johnston-linear-matrix-algebra_FO3950 | 422 | 0.996 | -i | | |
| johnston-linear-matrix-algebra_FO3951 | 422 | 1.000 | z=e^{\pi i / 2}=i | | |
| johnston-linear-matrix-algebra_FO3952 | 422 | 1.000 | z^{1 / 3}=e^{\pi i / 6}=(\sqrt{3}+i) / 2 | | |
| johnston-linear-matrix-algebra_FO3953 | 422 | 1.000 | p: \mathbb{R} \rightarrow \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO3954 | 422 | 1.000 | a_{0}, a_{1}, a_{2}, \ldots, a_{n-1}, a_{n} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO3955 | 422 | 0.995 | a_{n} \neq 0 | | |
| johnston-linear-matrix-algebra_FO3956 | 422 | 1.000 | p(x)=0 | | |
| johnston-linear-matrix-algebra_FO3957 | 422 | 1.000 | x=0.5 | | |
| johnston-linear-matrix-algebra_FO3958 | 422 | 1.000 | x=1.5 | | |
| johnston-linear-matrix-algebra_FO3959 | 423 | 1.000 | x^{2}-6 x+8 | | |
| johnston-linear-matrix-algebra_FO3960 | 423 | 1.000 | 2 x^{2}+8 x+8 | | |
| johnston-linear-matrix-algebra_FO3961 | 423 | 1.000 | x^{2}+2 x+3 | | |
| johnston-linear-matrix-algebra_FO3962 | 423 | 1.000 | a=1, b=-6 | | |
| johnston-linear-matrix-algebra_FO3963 | 423 | 1.000 | c=8 | | |
| johnston-linear-matrix-algebra_FO3964 | 423 | 1.000 | a=2, b=8 | | |
| johnston-linear-matrix-algebra_FO3965 | 423 | 1.000 | a=1, b=2 | | |
| johnston-linear-matrix-algebra_FO3966 | 423 | 1.000 | c=3 | | |
| johnston-linear-matrix-algebra_FO3967 | 424 | 1.000 | b / c | | |
| johnston-linear-matrix-algebra_FO3968 | 424 | 1.000 | -1 \pm i \sqrt{2} | | |
| johnston-linear-matrix-algebra_FO3969 | 424 | 1.000 | a_{0}, a_{n} \neq 0 | | |
| johnston-linear-matrix-algebra_FO3970 | 424 | 1.000 | a_{0} | | |
| johnston-linear-matrix-algebra_FO3971 | 424 | 1.000 | a_{n} | | |
| johnston-linear-matrix-algebra_FO3972 | 424 | 1.000 | x^{2}-2 x-3 | | |
| johnston-linear-matrix-algebra_FO3973 | 424 | 1.000 | 2 x^{3}-9 x^{2}-6 x+5 | | |
| johnston-linear-matrix-algebra_FO3974 | 424 | 1.000 | 3 x^{3}-14 x+4 | | |
| johnston-linear-matrix-algebra_FO3975 | 424 | 1.000 | a_{2}=1 | | |
| johnston-linear-matrix-algebra_FO3976 | 424 | 1.000 | a_{0}=3 | | |
| johnston-linear-matrix-algebra_FO3977 | 424 | 0.273 | \pm 1 | | |
| johnston-linear-matrix-algebra_FO3978 | 424 | 0.273 | \pm 3 | | |
| johnston-linear-matrix-algebra_FO3979 | 424 | 1.000 | x=-1 | | |
| johnston-linear-matrix-algebra_FO3980 | 424 | 1.000 | x=3 | | |
| johnston-linear-matrix-algebra_FO3981 | 425 | 1.000 | a_{3}=2 | | |
| johnston-linear-matrix-algebra_FO3982 | 425 | 1.000 | a_{0}=5 | | |
| johnston-linear-matrix-algebra_FO3983 | 425 | 0.967 | x=5, x=1 / 2 | | |
| johnston-linear-matrix-algebra_FO3984 | 425 | 1.000 | a_{3}=3 | | |
| johnston-linear-matrix-algebra_FO3985 | 425 | 1.000 | a_{0}=4 | | |
| johnston-linear-matrix-algebra_FO3986 | 425 | 0.880 | \pm 4 / 3, \pm 2 / 3 | | |
| johnston-linear-matrix-algebra_FO3987 | 425 | 0.880 | \pm 1 / 3 | | |
| johnston-linear-matrix-algebra_FO3988 | 425 | 1.000 | p(x)=3 x^{3}-14 x+4 | | |
| johnston-linear-matrix-algebra_FO3989 | 425 | 1.000 | \sqrt{2}, \pi | | |
| johnston-linear-matrix-algebra_FO3990 | 425 | 1.000 | p(x) | | |
| johnston-linear-matrix-algebra_FO3991 | 425 | 0.699 | \left(p(x)=3 x^{3}-14 x+4\right) | | |
| johnston-linear-matrix-algebra_FO3992 | 425 | 0.999 | (x-2) | | |
| johnston-linear-matrix-algebra_FO3993 | 425 | 1.000 | 3 x^{3} | | |
| johnston-linear-matrix-algebra_FO3994 | 425 | 1.000 | 3 x^{2} | | |
| johnston-linear-matrix-algebra_FO3995 | 426 | 0.999 | 0 x^{2}-\left(-6 x^{2}\right)=6 x^{2} | | |
| johnston-linear-matrix-algebra_FO3996 | 426 | 0.517 | x-2 | | |
| johnston-linear-matrix-algebra_FO3997 | 426 | 1.000 | 2\left(6 x^{2}-14 x+4\right) | | |
| johnston-linear-matrix-algebra_FO3998 | 426 | 0.933 | \left(q(x)=3 x^{2}+6 x-2\right) | | |
| johnston-linear-matrix-algebra_FO3999 | 426 | 1.000 | q(x)=3 x^{2}+6 x-2 | | |
| johnston-linear-matrix-algebra_FO4000 | 426 | 0.996 | 2,-1+\sqrt{5 / 3} | | |
| johnston-linear-matrix-algebra_FO4001 | 426 | 0.996 | -1-\sqrt{5 / 3} | | |
| johnston-linear-matrix-algebra_FO4002 | 426 | 0.997 | p(x)=3 x^{4}+4 x^{3}-14 x^{2}-11 x-2 | | |
| johnston-linear-matrix-algebra_FO4003 | 426 | 1.000 | a_{4}=3 | | |
| johnston-linear-matrix-algebra_FO4004 | 426 | 1.000 | a_{0}=-2 | | |
| johnston-linear-matrix-algebra_FO4005 | 426 | 1.000 | x=-1 / 3 | | |
| johnston-linear-matrix-algebra_FO4006 | 427 | 1.000 | x^{2}+3 x+1 | | |
| johnston-linear-matrix-algebra_FO4007 | 427 | 1.000 | 2, \frac{-1}{3}, \frac{1}{2}(-3+\sqrt{5}) | | |
| johnston-linear-matrix-algebra_FO4008 | 427 | 1.000 | \frac{1}{2}(-3-\sqrt{5}) | | |
| johnston-linear-matrix-algebra_FO4009 | 427 | 1.000 | x-r | | |
| johnston-linear-matrix-algebra_FO4010 | 427 | 1.000 | p(x)=(x-r) q(x) | | |
| johnston-linear-matrix-algebra_FO4011 | 428 | 1.000 | p(x)=x^{2}+1 | | |
| johnston-linear-matrix-algebra_FO4012 | 428 | 0.999 | x^{n} | | |
| johnston-linear-matrix-algebra_FO4013 | 428 | 0.999 | r_{1}, r_{2}, \ldots, r_{n} | | |
| johnston-linear-matrix-algebra_FO4014 | 428 | 0.679 | 2 x^{2}+8 x+8=2(x+2)^{2} | | |
| johnston-linear-matrix-algebra_FO4015 | 428 | 0.922 | x+2 | | |
| johnston-linear-matrix-algebra_FO4016 | 429 | 1.000 | x y=y x | | |
| johnston-linear-matrix-algebra_FO4017 | 429 | 1.000 | m^{2} | | |
| johnston-linear-matrix-algebra_FO4018 | 429 | 1.000 | m=2 k | | |
| johnston-linear-matrix-algebra_FO4019 | 429 | 1.000 | 2 k^{2} | | |
| johnston-linear-matrix-algebra_FO4020 | 429 | 1.000 | m=2 k+1 | | |
| johnston-linear-matrix-algebra_FO4021 | 429 | 1.000 | m=7 k | | |
| johnston-linear-matrix-algebra_FO4022 | 429 | 1.000 | m^{2}=(2 k)^{2} | | |
| johnston-linear-matrix-algebra_FO4023 | 429 | 1.000 | (2 k)^{2}=2\left(2 k^{2}\right) | | |
| johnston-linear-matrix-algebra_FO4024 | 429 | 1.000 | x^{2}+y^{2} \geq 2 x y | | |
| johnston-linear-matrix-algebra_FO4025 | 429 | 0.997 | x- | | |
| johnston-linear-matrix-algebra_FO4026 | 429 | 1.000 | y)^{2} \geq 0 | | |
| johnston-linear-matrix-algebra_FO4027 | 429 | 1.000 | (x-y)^{2} | | |
| johnston-linear-matrix-algebra_FO4028 | 429 | 1.000 | 2 x y | | |
| johnston-linear-matrix-algebra_FO4029 | 429 | 1.000 | 2 x y \leq x^{2}+y^{2} | | |
| johnston-linear-matrix-algebra_FO4030 | 430 | 0.889 | x^{2}+y^{2}-2 x y | | |
| johnston-linear-matrix-algebra_FO4031 | 430 | 0.998 | a, b, c | | |
| johnston-linear-matrix-algebra_FO4032 | 430 | 0.998 | a x^{2}+b x+c=0 | | |
| johnston-linear-matrix-algebra_FO4033 | 430 | 1.000 | b x | | |
| johnston-linear-matrix-algebra_FO4034 | 430 | 1.000 | a x^{2} | | |
| johnston-linear-matrix-algebra_FO4035 | 431 | 1.000 | m^{2}=2 k | | |
| johnston-linear-matrix-algebra_FO4036 | 431 | 1.000 | Q^{\prime \prime} | | |
| johnston-linear-matrix-algebra_FO4037 | 431 | 1.000 | 2\left(2 k^{2}+2 k\right) | | |
| johnston-linear-matrix-algebra_FO4038 | 431 | 0.597 | 2 \times 1 | | |
| johnston-linear-matrix-algebra_FO4039 | 432 | 0.997 | 8 \times 8 | | |
| johnston-linear-matrix-algebra_FO4040 | 432 | 0.666 | 8 \times 8=64 | | |
| johnston-linear-matrix-algebra_FO4041 | 434 | 1.000 | \sqrt{2}=a / b | | |
| johnston-linear-matrix-algebra_FO4042 | 434 | 1.000 | a / b | | |
| johnston-linear-matrix-algebra_FO4043 | 434 | 1.000 | \sqrt{2} b=a | | |
| johnston-linear-matrix-algebra_FO4044 | 434 | 1.000 | 2 b^{2}=a^{2} | | |
| johnston-linear-matrix-algebra_FO4045 | 434 | 1.000 | a^{2} | | |
| johnston-linear-matrix-algebra_FO4046 | 434 | 1.000 | a=2 k | | |
| johnston-linear-matrix-algebra_FO4047 | 435 | 1.000 | 2 b^{2}=(2 k)^{2}=4 k^{2} | | |
| johnston-linear-matrix-algebra_FO4048 | 435 | 1.000 | b^{2}=2 k^{2} | | |
| johnston-linear-matrix-algebra_FO4049 | 435 | 1.000 | b^{2} | | |
| johnston-linear-matrix-algebra_FO4050 | 435 | 1.000 | 1+3+5+\cdots+(2 n-1)=n^{2} | | |
| johnston-linear-matrix-algebra_FO4051 | 435 | 1.000 | n \leq 5 | | |
| johnston-linear-matrix-algebra_FO4052 | 435 | 0.999 | P(1), P(2), P(3), \ldots | | |
| johnston-linear-matrix-algebra_FO4053 | 435 | 0.999 | 1,2,3, \ldots | | |
| johnston-linear-matrix-algebra_FO4054 | 435 | 1.000 | P(1) | | |
| johnston-linear-matrix-algebra_FO4055 | 435 | 1.000 | P(n) | | |
| johnston-linear-matrix-algebra_FO4056 | 435 | 1.000 | P(n+1) | | |
| johnston-linear-matrix-algebra_FO4057 | 435 | 1.000 | P(2) | | |
| johnston-linear-matrix-algebra_FO4058 | 435 | 1.000 | P(3) | | |
| johnston-linear-matrix-algebra_FO4059 | 435 | 0.988 | P(4) | | |
| johnston-linear-matrix-algebra_FO4060 | 435 | 1.000 | 1=1 | | |
| johnston-linear-matrix-algebra_FO4061 | 436 | 1.000 | 2 n+1 | | |
| johnston-linear-matrix-algebra_FO4062 | 436 | 0.995 | 1,3,5, \ldots, 2 n-1 | | |
| johnston-linear-matrix-algebra_FO4063 | 437 | 0.992 | \left[A_{i, j}\right]_{k, \ell} | | |
| johnston-linear-matrix-algebra_FO4064 | 437 | 0.992 | (k, \ell) | | |
| johnston-linear-matrix-algebra_FO4065 | 437 | 1.000 | A_{i, j} | | |
| johnston-linear-matrix-algebra_FO4066 | 438 | 1.000 | R, S \in \mathcal{M}_{m, n} | | |
| johnston-linear-matrix-algebra_FO4067 | 438 | 1.000 | R=S | | |
| johnston-linear-matrix-algebra_FO4068 | 438 | 1.000 | R \neq S | | |
| johnston-linear-matrix-algebra_FO4069 | 438 | 1.000 | \widetilde{R} | | |
| johnston-linear-matrix-algebra_FO4070 | 438 | 1.000 | \widetilde{S} | | |
| johnston-linear-matrix-algebra_FO4071 | 439 | 1.000 | \widetilde{\mathbf{R}} | | |
| johnston-linear-matrix-algebra_FO4072 | 439 | 0.960 | \mathbf{c}=\mathbf{b} | | |
| johnston-linear-matrix-algebra_FO4073 | 439 | 1.000 | \widetilde{S}=\widetilde{R} | | |
| johnston-linear-matrix-algebra_FO4074 | 439 | 1.000 | \widetilde{R}=\widetilde{S} | | |
| johnston-linear-matrix-algebra_FO4075 | 439 | 1.000 | I_{m} | | |
| johnston-linear-matrix-algebra_FO4076 | 439 | 1.000 | E A=B | | |
| johnston-linear-matrix-algebra_FO4077 | 439 | 0.976 | (j, j) | | |
| johnston-linear-matrix-algebra_FO4078 | 440 | 1.000 | \mathbf{e}_{j}+c \mathbf{e}_{i} | | |
| johnston-linear-matrix-algebra_FO4079 | 440 | 1.000 | \mathbf{e}_{i}+c \mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO4080 | 440 | 1.000 | \mathbf{e}_{1} \mathbf{a}_{1}^{T} | | |
| johnston-linear-matrix-algebra_FO4081 | 440 | 1.000 | \mathbf{a}_{1}^{T} | | |
| johnston-linear-matrix-algebra_FO4082 | 440 | 1.000 | \{1,2, \ldots, n\}, \sigma | | |
| johnston-linear-matrix-algebra_FO4083 | 440 | 1.000 | \mathbf{e}_{i} \mathbf{a}_{j}^{T} | | |
| johnston-linear-matrix-algebra_FO4084 | 440 | 1.000 | A+c \mathbf{e}_{i} \mathbf{a}_{j}^{T} | | |
| johnston-linear-matrix-algebra_FO4085 | 441 | 1.000 | v+c \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4086 | 441 | 0.996 | \sigma(j)=j | | |
| johnston-linear-matrix-algebra_FO4087 | 442 | 1.000 | p_{A}^{\prime} | | |
| johnston-linear-matrix-algebra_FO4088 | 442 | 1.000 | \frac{\partial}{\partial \lambda_{i}} | | |
| johnston-linear-matrix-algebra_FO4089 | 442 | 1.000 | \lambda_{i} | | |
| johnston-linear-matrix-algebra_FO4090 | 442 | 1.000 | \mu \neq 0 | | |
| johnston-linear-matrix-algebra_FO4091 | 442 | 1.000 | L(|\mathbf{w}|)=\mu | | |
| johnston-linear-matrix-algebra_FO4092 | 442 | 1.000 | A|\mathbf{w}|=\mu|\mathbf{w}| | | |
| johnston-linear-matrix-algebra_FO4093 | 442 | 1.000 | |\mathbf{w}|>\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4094 | 442 | 0.948 | (A-B)|\mathbf{w}|=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4095 | 442 | 1.000 | A-B=O | | |
| johnston-linear-matrix-algebra_FO4096 | 442 | 1.000 | p_{A}(\lambda)=\operatorname{det}(A- | | |
| johnston-linear-matrix-algebra_FO4097 | 442 | 0.986 | \lambda I) | | |
| johnston-linear-matrix-algebra_FO4098 | 442 | 1.000 | p_{A}^{\prime}(\mu) \neq 0 | | |
| johnston-linear-matrix-algebra_FO4099 | 442 | 1.000 | f: \mathbb{R}^{n} \rightarrow \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO4100 | 442 | 0.999 | \Lambda | | |
| johnston-linear-matrix-algebra_FO4101 | 442 | 1.000 | A_{i} | | |
| johnston-linear-matrix-algebra_FO4102 | 442 | 1.000 | \Lambda_{i} | | |
| johnston-linear-matrix-algebra_FO4103 | 442 | 1.000 | A-\Lambda | | |
| johnston-linear-matrix-algebra_FO4104 | 442 | 1.000 | a_{i, 1}, \ldots, a_{i, n} | | |
| johnston-linear-matrix-algebra_FO4105 | 442 | 1.000 | c_{i, 1}, \ldots, c_{i, n} | | |
| johnston-linear-matrix-algebra_FO4106 | 442 | 1.000 | B_{i} | | |
| johnston-linear-matrix-algebra_FO4107 | 442 | 1.000 | p_{B_{i}}(\lambda)=-\lambda p_{A}(\lambda) | | |
| johnston-linear-matrix-algebra_FO4108 | 442 | 1.000 | O \leq B_{i} \leq A | | |
| johnston-linear-matrix-algebra_FO4109 | 442 | 1.000 | p_{B_{i}}(\mu)>0 | | |
| johnston-linear-matrix-algebra_FO4110 | 442 | 1.000 | p_{B_{i}}(\mu)<0 | | |
| johnston-linear-matrix-algebra_FO4111 | 443 | 1.000 | p^{(k)} | | |
| johnston-linear-matrix-algebra_FO4112 | 443 | 0.999 | (f g)^{\prime}(x)= | | |
| johnston-linear-matrix-algebra_FO4113 | 443 | 0.715 | f^{\prime}(x) g(x)+f(x) g^{\prime}(x) | | |
| johnston-linear-matrix-algebra_FO4114 | 443 | 1.000 | m-1 | | |
| johnston-linear-matrix-algebra_FO4115 | 443 | 1.000 | p, p^{\prime}, \ldots, p^{(m-1)} | | |
| johnston-linear-matrix-algebra_FO4116 | 443 | 1.000 | p^{(m)} | | |
| johnston-linear-matrix-algebra_FO4117 | 443 | 0.981 | p(x)= | | |
| johnston-linear-matrix-algebra_FO4118 | 443 | 1.000 | (x-r)^{m} q(x) | | |
| johnston-linear-matrix-algebra_FO4119 | 443 | 1.000 | q(r) \neq 0 | | |
| johnston-linear-matrix-algebra_FO4120 | 443 | 1.000 | c_{k} \neq 0 | | |
| johnston-linear-matrix-algebra_FO4121 | 443 | 1.000 | s_{k}(x) | | |
| johnston-linear-matrix-algebra_FO4122 | 443 | 1.000 | s_{k}(r)=0 | | |
| johnston-linear-matrix-algebra_FO4123 | 443 | 1.000 | c_{0}=1 | | |
| johnston-linear-matrix-algebra_FO4124 | 443 | 0.999 | s_{k}(x)=0 | | |
| johnston-linear-matrix-algebra_FO4125 | 443 | 1.000 | c_{1}=m | | |
| johnston-linear-matrix-algebra_FO4126 | 443 | 1.000 | s_{1}(x)= | | |
| johnston-linear-matrix-algebra_FO4127 | 443 | 1.000 | (x-r) q^{\prime}(x) | | |
| johnston-linear-matrix-algebra_FO4128 | 443 | 1.000 | p^{(k+1)} | | |
| johnston-linear-matrix-algebra_FO4129 | 443 | 1.000 | 0 \leq k \leq m | | |
| johnston-linear-matrix-algebra_FO4130 | 443 | 1.000 | x=r | | |
| johnston-linear-matrix-algebra_FO4131 | 443 | 1.000 | p^{(k)}(r)=0 | | |
| johnston-linear-matrix-algebra_FO4132 | 443 | 1.000 | 0 \leq k<m | | |
| johnston-linear-matrix-algebra_FO4133 | 443 | 1.000 | s_{m}(r)=0 | | |
| johnston-linear-matrix-algebra_FO4134 | 443 | 1.000 | c_{m}, q(r) \neq 0 | | |
| johnston-linear-matrix-algebra_FO4135 | 444 | 1.000 | n x^{n-1} | | |
| johnston-linear-matrix-algebra_FO4136 | 444 | 1.000 | p_{0}, p_{1}, \ldots, p_{m} | | |
| johnston-linear-matrix-algebra_FO4137 | 444 | 1.000 | r \neq 0 | | |
| johnston-linear-matrix-algebra_FO4138 | 444 | 1.000 | p_{0} | | |
| johnston-linear-matrix-algebra_FO4139 | 444 | 1.000 | m=0 | | |
| johnston-linear-matrix-algebra_FO4140 | 444 | 1.000 | m+1=2 | | |
| johnston-linear-matrix-algebra_FO4141 | 444 | 1.000 | p_{0}^{\prime} | | |
| johnston-linear-matrix-algebra_FO4142 | 444 | 1.000 | p_{1}(x)=x p_{0}^{\prime}(x) | | |
| johnston-linear-matrix-algebra_FO4143 | 444 | 1.000 | p_{0}^{\prime}(x) | | |
| johnston-linear-matrix-algebra_FO4144 | 444 | 1.000 | m>1 | | |
| johnston-linear-matrix-algebra_FO4145 | 444 | 1.000 | p_{2}(x)=x p_{1}^{\prime}(x), p_{3}(x)=x p_{2}^{\prime}(x) | | |
| johnston-linear-matrix-algebra_FO4146 | 444 | 0.931 | 0 / 0 | | |
| johnston-linear-matrix-algebra_FO4147 | 444 | 0.931 | \infty / \infty | | |
| johnston-linear-matrix-algebra_FO4148 | 444 | 1.000 | q(x)=c_{k} x^{k}+\cdots+c_{1} x+c_{0} | | |
| johnston-linear-matrix-algebra_FO4149 | 445 | 1.000 | b^{x} | | |
| johnston-linear-matrix-algebra_FO4150 | 445 | 1.000 | b^{x} \ln (b) | | |
| johnston-linear-matrix-algebra_FO4151 | 445 | 0.996 | c_{k} k! | | |
| johnston-linear-matrix-algebra_FO4152 | 445 | 1.000 | b=1 / c | | |
| johnston-linear-matrix-algebra_FO4153 | 445 | 1.000 | |b|>1 | | |
| johnston-linear-matrix-algebra_FO4154 | 446 | 1.000 | \mathbf{v}=2 \mathbf{e}_{3} | | |
| johnston-linear-matrix-algebra_FO4155 | 446 | 0.992 | \mathbf{x}=\mathbf{e}_{1}+2 \mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO4156 | 446 | 1.000 | c(1,2)=(3,4) | | |
| johnston-linear-matrix-algebra_FO4157 | 446 | 1.000 | 2 c=4 | | |
| johnston-linear-matrix-algebra_FO4158 | 446 | 1.000 | c=2 | | |
| johnston-linear-matrix-algebra_FO4159 | 446 | 0.990 | (3,4) | | |
| johnston-linear-matrix-algebra_FO4160 | 446 | 1.000 | \mathbf{v}+\mathbf{w}=\mathbf{x} | | |
| johnston-linear-matrix-algebra_FO4161 | 446 | 1.000 | \mathbf{v}-\mathbf{w}=\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4162 | 446 | 1.000 | 2 \mathbf{v}=\mathbf{x}+\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4163 | 446 | 1.000 | 2 \mathbf{w}=\mathbf{x}-\mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4164 | 446 | 1.000 | \mathbf{v}=(\mathbf{x}+\mathbf{y}) / 2=(2,1) | | |
| johnston-linear-matrix-algebra_FO4165 | 446 | 1.000 | \mathbf{w}=(\mathbf{x}-\mathbf{y}) / 2=(1,-3) | | |
| johnston-linear-matrix-algebra_FO4166 | 446 | 0.996 | \mathbf{x}=(2,2) | | |
| johnston-linear-matrix-algebra_FO4167 | 446 | 1.000 | \mathbf{x}=\frac{1}{2}(\mathbf{v}-\mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO4168 | 447 | 1.000 | \|\mathbf{v}\|=5, \mathbf{u}=(3 / 5,4 / 5) | | |
| johnston-linear-matrix-algebra_FO4169 | 447 | 1.000 | \|\mathbf{v}\|=6, \mathbf{u}=(-\sqrt{2} / 3,-1 / 2, \sqrt{10} / 6,1 / 2) | | |
| johnston-linear-matrix-algebra_FO4170 | 447 | 1.000 | \arccos (-1 / \sqrt{2})=3 \pi / 4 | | |
| johnston-linear-matrix-algebra_FO4171 | 447 | 1.000 | \mathbf{x}=(1,-1) | | |
| johnston-linear-matrix-algebra_FO4172 | 447 | 1.000 | \mathbf{v} \cdot \mathbf{x}=1 | | |
| johnston-linear-matrix-algebra_FO4173 | 447 | 1.000 | \| \mathbf{v}+ | | |
| johnston-linear-matrix-algebra_FO4174 | 447 | 0.942 | \mathbf{w}\|\leq\| \mathbf{v}\|+\| \mathbf{w} \| \leq 2 | | |
| johnston-linear-matrix-algebra_FO4175 | 447 | 0.995 | (1,0,0) | | |
| johnston-linear-matrix-algebra_FO4176 | 447 | 0.995 | \mathbf{w}=(-1, \sqrt{3}, 0) | | |
| johnston-linear-matrix-algebra_FO4177 | 447 | 1.000 | \mathbf{v}=(2,0) | | |
| johnston-linear-matrix-algebra_FO4178 | 447 | 1.000 | \mathbf{w}=(0,2) | | |
| johnston-linear-matrix-algebra_FO4179 | 447 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\cos (\theta)\|\mathbf{v}\|\|\mathbf{w}\|=\cos (\pi / 3)(2 \sqrt{3})(2)= | | |
| johnston-linear-matrix-algebra_FO4180 | 447 | 1.000 | 2 \sqrt{3} | | |
| johnston-linear-matrix-algebra_FO4181 | 447 | 1.000 | 3 w_{1}+\sqrt{3} w_{2}=2 \sqrt{3} | | |
| johnston-linear-matrix-algebra_FO4182 | 447 | 1.000 | w_{2}=2-\sqrt{3} w_{1} | | |
| johnston-linear-matrix-algebra_FO4183 | 447 | 1.000 | w_{1}^{2}+w_{2}^{2}=4 | | |
| johnston-linear-matrix-algebra_FO4184 | 447 | 1.000 | w_{2} | | |
| johnston-linear-matrix-algebra_FO4185 | 447 | 1.000 | w_{1}^{2}+\left(2-\sqrt{3} w_{1}\right)^{2}=4 | | |
| johnston-linear-matrix-algebra_FO4186 | 447 | 1.000 | w_{1}\left(w_{1}-\sqrt{3}\right)=0 | | |
| johnston-linear-matrix-algebra_FO4187 | 447 | 1.000 | w_{1}=0 | | |
| johnston-linear-matrix-algebra_FO4188 | 447 | 1.000 | w_{1}=\sqrt{3} | | |
| johnston-linear-matrix-algebra_FO4189 | 447 | 0.999 | w_{2}=2 | | |
| johnston-linear-matrix-algebra_FO4190 | 447 | 0.999 | w_{2}=-1 | | |
| johnston-linear-matrix-algebra_FO4191 | 447 | 0.925 | \mathbf{w}=(\sqrt{3},-1) | | |
| johnston-linear-matrix-algebra_FO4192 | 447 | 1.000 | \mathbf{v}^{2} | | |
| johnston-linear-matrix-algebra_FO4193 | 447 | 0.974 | \|\mathbf{v}\|^{2}=\mathbf{v} \cdot \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4194 | 447 | 0.996 | \mathbf{w}=\left(w_{1}, w_{2}\right) \in \mathbb{R}^{2} | | |
| johnston-linear-matrix-algebra_FO4195 | 447 | 0.996 | w_{1}+2 w_{2}=0 | | |
| johnston-linear-matrix-algebra_FO4196 | 447 | 1.000 | w_{1}= | | |
| johnston-linear-matrix-algebra_FO4197 | 447 | 1.000 | -2 w_{2} | | |
| johnston-linear-matrix-algebra_FO4198 | 447 | 0.975 | \mathbf{v} \cdot \mathbf{y}=0 | | |
| johnston-linear-matrix-algebra_FO4199 | 447 | 0.531 | \mathbf{y}=y_{2}(-2,1) | | |
| johnston-linear-matrix-algebra_FO4200 | 447 | 1.000 | \mathbf{w} \cdot \mathbf{y}=y_{2}(4+1) | | |
| johnston-linear-matrix-algebra_FO4201 | 447 | 1.000 | \mathbf{y}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4202 | 447 | 1.000 | (2,2,2)+(-1,-2,-3)+(-3,0,1)=(-2,0,0) | | |
| johnston-linear-matrix-algebra_FO4203 | 447 | 1.000 | \overline{\bar{x}}=x | | |
| johnston-linear-matrix-algebra_FO4204 | 447 | 1.000 | \overline{x \cdot y}= | | |
| johnston-linear-matrix-algebra_FO4205 | 447 | 1.000 | \bar{x} \cdot \bar{y} | | |
| johnston-linear-matrix-algebra_FO4206 | 447 | 1.000 | x, y \in \mathbb{C} | | |
| johnston-linear-matrix-algebra_FO4207 | 447 | 1.000 | \|\mathbf{v}\|\|\mathbf{w}\|=0 | | |
| johnston-linear-matrix-algebra_FO4208 | 447 | 0.992 | |\mathbf{v} \cdot \mathbf{w}|=|(c \mathbf{w}) \cdot \mathbf{w}|=|c|\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO4209 | 447 | 1.000 | \|\mathbf{v}\|\|\mathbf{w}\|=\|c \mathbf{w}\|\|\mathbf{w}\|=|c|\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO4210 | 447 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO4211 | 447 | 0.985 | \|\mathbf{w}\| \mathbf{v}-\|\mathbf{v}\| \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4212 | 447 | 0.555 | \|\mathbf{w}\| \mathbf{v}=\|\mathbf{v}\| \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4213 | 448 | 1.000 | \|\mathbf{v}\|=\|\mathbf{v}\| | | |
| johnston-linear-matrix-algebra_FO4214 | 448 | 1.000 | \|\mathbf{v}+\mathbf{w}\|=\|(c \mathbf{w})+\mathbf{w}\|= | | |
| johnston-linear-matrix-algebra_FO4215 | 448 | 1.000 | \|(c+1) \mathbf{w}\|=(c+1)\|\mathbf{w}\|=c\|\mathbf{w}\|+\|\mathbf{w}\|= | | |
| johnston-linear-matrix-algebra_FO4216 | 448 | 1.000 | \|c \mathbf{w}\|+\|\mathbf{w}\|=\|\mathbf{v}\|+\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO4217 | 448 | 1.000 | \|\mathbf{v}+\mathbf{w}\|= | | |
| johnston-linear-matrix-algebra_FO4218 | 448 | 1.000 | 2(\mathbf{v} \cdot \mathbf{w})=2\|\mathbf{v}\|\|\mathbf{w}\| | | |
| johnston-linear-matrix-algebra_FO4219 | 448 | 1.000 | c \geq 0 | | |
| johnston-linear-matrix-algebra_FO4220 | 448 | 1.000 | 2(\mathbf{v} \cdot \mathbf{w})=2 c\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO4221 | 448 | 0.956 | 2\|\mathbf{v}\|\|\mathbf{w}\|=2|c|\|\mathbf{w}\|^{2}=-2 c\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO4222 | 448 | 1.000 | \|\mathbf{x}+\mathbf{y}\| \leq\|\mathbf{x}\|+ | | |
| johnston-linear-matrix-algebra_FO4223 | 448 | 1.000 | \|\mathbf{y}\| | | |
| johnston-linear-matrix-algebra_FO4224 | 448 | 1.000 | \mathbf{x}=\mathbf{v}-\mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4225 | 448 | 1.000 | \mathbf{y}=\mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4226 | 448 | 0.999 | f(x)=\|\mathbf{v}-x \mathbf{w}\|^{2}=(\mathbf{v}-x \mathbf{w}) \cdot(\mathbf{v}-x \mathbf{w})= | | |
| johnston-linear-matrix-algebra_FO4227 | 448 | 1.000 | x^{2}\|\mathbf{w}\|^{2}-2 x(\mathbf{v} \cdot \mathbf{w})+\|\mathbf{v}\|^{2} | | |
| johnston-linear-matrix-algebra_FO4228 | 448 | 1.000 | 4(\mathbf{v} \cdot \mathbf{w})^{2}-4\|\mathbf{v}\|^{2}\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO4229 | 448 | 1.000 | 4(\mathbf{v} \cdot \mathbf{w})^{2}- | | |
| johnston-linear-matrix-algebra_FO4230 | 448 | 1.000 | 4\|\mathbf{v}\|^{2}\|\mathbf{w}\|^{2} \leq 0 | | |
| johnston-linear-matrix-algebra_FO4231 | 448 | 1.000 | 4\|\mathbf{v}\|^{2}\|\mathbf{w}\|^{2} | | |
| johnston-linear-matrix-algebra_FO4232 | 448 | 1.000 | \left[\begin{array}{cc}1 & 0 \\ -2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4233 | 448 | 1.000 | \left[\begin{array}{cc}3 & -2 \\ 2 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4234 | 448 | 1.000 | A^{5}=A^{1}, A^{6}=A^{2} | | |
| johnston-linear-matrix-algebra_FO4235 | 448 | 0.979 | A^{2}=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4236 | 448 | 1.000 | A^{3}=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4237 | 449 | 1.000 | k=0,1 | | |
| johnston-linear-matrix-algebra_FO4238 | 449 | 1.000 | k=2,3 | | |
| johnston-linear-matrix-algebra_FO4239 | 449 | 0.808 | A^{\ell}=\left[\begin{array}{cc}1 & \ell \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4240 | 449 | 1.000 | \left[J_{n}\right]_{i, j}=(1,1, \ldots, 1) \cdot(1,1, \ldots, 1)=n | | |
| johnston-linear-matrix-algebra_FO4241 | 449 | 1.000 | J_{n}^{2}=n J_{n} | | |
| johnston-linear-matrix-algebra_FO4242 | 449 | 1.000 | A C^{T} | | |
| johnston-linear-matrix-algebra_FO4243 | 449 | 1.000 | A B=\left[\begin{array}{ll|ll}3 & 3 & 3 & 3 \\ 3 & 3 & 3 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4244 | 449 | 0.997 | A B=\left[\begin{array}{rr|r}2 & 4 & 6 \\ 3 & 5 & 7 \\ 4 & 6 & 8 \\ \hline 2 & 4 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4245 | 449 | 1.000 | r=n | | |
| johnston-linear-matrix-algebra_FO4246 | 449 | 1.000 | p=m | | |
| johnston-linear-matrix-algebra_FO4247 | 449 | 1.000 | m=n=r=p | | |
| johnston-linear-matrix-algebra_FO4248 | 449 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\mathbf{v}^{T} \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4249 | 449 | 1.000 | \mathbf{x} \cdot(A \mathbf{y})= | | |
| johnston-linear-matrix-algebra_FO4250 | 449 | 1.000 | \mathbf{x}^{T} A \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4251 | 449 | 1.000 | \left(A^{T} \mathbf{x}\right) \cdot \mathbf{y}=\left(A^{T} \mathbf{x}\right)^{T} \mathbf{y}=\mathbf{x}^{T} A \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4252 | 449 | 1.000 | \mathbf{x} \cdot(A \mathbf{y})=\mathbf{x}^{T} A \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4253 | 449 | 1.000 | (B \mathbf{x}) \cdot \mathbf{y}=\mathbf{x}^{T} B^{T} \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4254 | 449 | 1.000 | \mathbf{x}^{T} A \mathbf{y}= | | |
| johnston-linear-matrix-algebra_FO4255 | 449 | 1.000 | \mathbf{x}^{T} B^{T} \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4256 | 449 | 1.000 | \mathbf{x}=\mathbf{e}_{i} | | |
| johnston-linear-matrix-algebra_FO4257 | 449 | 1.000 | \mathbf{y}=\mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO4258 | 449 | 1.000 | \mathbf{x}^{T} A \mathbf{y}=a_{i, j} | | |
| johnston-linear-matrix-algebra_FO4259 | 449 | 1.000 | \mathbf{x}^{T} B^{T} \mathbf{y}=b_{j, i} | | |
| johnston-linear-matrix-algebra_FO4260 | 449 | 1.000 | a_{i, j}=b_{j, i} | | |
| johnston-linear-matrix-algebra_FO4261 | 449 | 0.994 | \left[\mathbf{e}_{i}\right]_{k}=0 | | |
| johnston-linear-matrix-algebra_FO4262 | 449 | 0.994 | k \neq i | | |
| johnston-linear-matrix-algebra_FO4263 | 449 | 0.994 | \left[\mathbf{e}_{i}\right]_{i}=1 | | |
| johnston-linear-matrix-algebra_FO4264 | 449 | 1.000 | \mathbf{a}_{i} | | |
| johnston-linear-matrix-algebra_FO4265 | 449 | 1.000 | (A-B) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4266 | 449 | 1.000 | (A-B) \mathbf{e}_{i}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4267 | 449 | 1.000 | (A+B)+C | | |
| johnston-linear-matrix-algebra_FO4268 | 449 | 1.000 | A+(B+C) | | |
| johnston-linear-matrix-algebra_FO4269 | 449 | 1.000 | (A+B)+C= | | |
| johnston-linear-matrix-algebra_FO4270 | 449 | 1.000 | c(A+B) | | |
| johnston-linear-matrix-algebra_FO4271 | 449 | 1.000 | c A+c B | | |
| johnston-linear-matrix-algebra_FO4272 | 449 | 0.987 | c A+ | | |
| johnston-linear-matrix-algebra_FO4273 | 449 | 1.000 | c B | | |
| johnston-linear-matrix-algebra_FO4274 | 450 | 1.000 | (c+d) A | | |
| johnston-linear-matrix-algebra_FO4275 | 450 | 1.000 | c A+d A | | |
| johnston-linear-matrix-algebra_FO4276 | 450 | 1.000 | d A | | |
| johnston-linear-matrix-algebra_FO4277 | 450 | 1.000 | c(d A) | | |
| johnston-linear-matrix-algebra_FO4278 | 450 | 0.936 | (c d) A | | |
| johnston-linear-matrix-algebra_FO4279 | 450 | 1.000 | (A B) C | | |
| johnston-linear-matrix-algebra_FO4280 | 450 | 1.000 | A(B C) | | |
| johnston-linear-matrix-algebra_FO4281 | 450 | 1.000 | A C+B C | | |
| johnston-linear-matrix-algebra_FO4282 | 450 | 1.000 | A(B+C)= | | |
| johnston-linear-matrix-algebra_FO4283 | 450 | 1.000 | c(A B) | | |
| johnston-linear-matrix-algebra_FO4284 | 450 | 1.000 | (c A) B | | |
| johnston-linear-matrix-algebra_FO4285 | 450 | 0.734 | I_{m} A | | |
| johnston-linear-matrix-algebra_FO4286 | 450 | 0.998 | A: a_{i, j} | | |
| johnston-linear-matrix-algebra_FO4287 | 450 | 1.000 | A^{k} A^{\ell}=(A A \cdots A)(A A \cdots A) | | |
| johnston-linear-matrix-algebra_FO4288 | 450 | 1.000 | k+\ell | | |
| johnston-linear-matrix-algebra_FO4289 | 450 | 1.000 | A^{k+\ell} | | |
| johnston-linear-matrix-algebra_FO4290 | 450 | 1.000 | k \ell | | |
| johnston-linear-matrix-algebra_FO4291 | 450 | 1.000 | A^{k \ell} | | |
| johnston-linear-matrix-algebra_FO4292 | 450 | 1.000 | \left[\left(A^{T}\right)^{T}\right]_{i, j}=\left[A^{T}\right]_{j, i}=[A]_{i, j} | | |
| johnston-linear-matrix-algebra_FO4293 | 450 | 1.000 | [(A+ | | |
| johnston-linear-matrix-algebra_FO4294 | 450 | 0.431 | \left.{ }^{T}\right]_{i, j}=[A+B]_{j, i}=[A]_{j, i}+[B]_{j, i}=\left[A^{T}\right]_{i, j}+ | | |
| johnston-linear-matrix-algebra_FO4295 | 450 | 1.000 | \left[B^{T}\right]_{i, j}=\left[A^{T}+B^{T}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO4296 | 450 | 1.000 | \left[(c A)^{T}\right]_{i, j}=[c A]_{j, i}=c[A]_{j, i}=c\left[A^{T}\right]_{i, j}= | | |
| johnston-linear-matrix-algebra_FO4297 | 450 | 0.998 | \left[c A^{T}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO4298 | 450 | 1.000 | p \times 1 | | |
| johnston-linear-matrix-algebra_FO4299 | 450 | 1.000 | \mathbf{a}_{1}, \mathbf{a}_{2}, \ldots \mathbf{a}_{n} | | |
| johnston-linear-matrix-algebra_FO4300 | 450 | 1.000 | (A B)^{T}= | | |
| johnston-linear-matrix-algebra_FO4301 | 450 | 1.000 | \left[\begin{array}{cc}1 & 2 \\ 3 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4302 | 450 | 1.000 | \left[\begin{array}{ccc}1 & 1 & 0 \\ 1 & 1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4303 | 451 | 1.000 | T(1,0,0)= | | |
| johnston-linear-matrix-algebra_FO4304 | 451 | 1.000 | T(0,1,0)=T(1,1,0)-T(1,0,0)= | | |
| johnston-linear-matrix-algebra_FO4305 | 451 | 1.000 | (0,1,2)-(1,2,3)=(-1,-1,-1) | | |
| johnston-linear-matrix-algebra_FO4306 | 451 | 1.000 | T(0,0,1)=T(1,1,1)-T(1,1,0)= | | |
| johnston-linear-matrix-algebra_FO4307 | 451 | 1.000 | (0,0,1)-(0,1,2)=(0,-1,-1) | | |
| johnston-linear-matrix-algebra_FO4308 | 451 | 1.000 | [T]= | | |
| johnston-linear-matrix-algebra_FO4309 | 451 | 1.000 | \left[T\left(\mathbf{e}_{1}\right) \mid T\left(\mathbf{e}_{2}\right)\right] | | |
| johnston-linear-matrix-algebra_FO4310 | 451 | 1.000 | T\left(\mathbf{e}_{1}\right)=(2,1) | | |
| johnston-linear-matrix-algebra_FO4311 | 451 | 1.000 | T\left(\mathbf{e}_{2}\right)=(1,3) | | |
| johnston-linear-matrix-algebra_FO4312 | 451 | 1.000 | T(1,1)=T\left(\mathbf{e}_{1}+\mathbf{e}_{2}\right)=T\left(\mathbf{e}_{1}\right)+T\left(\mathbf{e}_{2}\right)= | | |
| johnston-linear-matrix-algebra_FO4313 | 451 | 1.000 | (2,1)+(1,3)=(3,4) \neq(3,3) | | |
| johnston-linear-matrix-algebra_FO4314 | 451 | 1.000 | \mathbf{v}=(0,1,0) | | |
| johnston-linear-matrix-algebra_FO4315 | 451 | 1.000 | R_{x z}^{\theta}(\mathbf{v})=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4316 | 451 | 0.623 | \left(R_{x y}^{\theta} \circ R_{y z}^{\theta}\right)(\mathbf{v})=R_{x y}^{\theta}(0,0,1)=(0,0,1) \neq \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4317 | 451 | 1.000 | R_{x y}^{\theta} \circ R_{y z}^{\theta} \neq R_{x z}^{\theta} | | |
| johnston-linear-matrix-algebra_FO4318 | 451 | 1.000 | T(2(1,1))=(4,2) | | |
| johnston-linear-matrix-algebra_FO4319 | 451 | 1.000 | 2 T(1,1)=(2,2) | | |
| johnston-linear-matrix-algebra_FO4320 | 451 | 0.996 | T(0,0)=(0,-1) \neq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4321 | 451 | 1.000 | T(2(1,1))=(2 \sqrt{2}, 2) | | |
| johnston-linear-matrix-algebra_FO4322 | 451 | 1.000 | 2 T(1,1)= | | |
| johnston-linear-matrix-algebra_FO4323 | 451 | 1.000 | (4,2 \sqrt{2}) | | |
| johnston-linear-matrix-algebra_FO4324 | 451 | 1.000 | T(1,1)+T(-1,-1)=(2,2) | | |
| johnston-linear-matrix-algebra_FO4325 | 451 | 1.000 | T((1,1)+(-1,-1))=(0,0) | | |
| johnston-linear-matrix-algebra_FO4326 | 451 | 1.000 | \frac{1}{10}\left[\begin{array}{ll}1 & 3 \\ 3 & 9\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4327 | 451 | 0.999 | \frac{1}{5}\left[\begin{array}{cc}-3 & 4 \\ 4 & 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4328 | 451 | 0.798 | \left[\begin{array}{cc}\cos (\pi / 5) & -\sin (\pi / 5) \\ \sin (\pi / 5) & \cos (\pi / 5)\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4329 | 451 | 1.000 | \mathbf{u}=\frac{1}{\sqrt{2}}(1,1) | | |
| johnston-linear-matrix-algebra_FO4330 | 451 | 0.902 | T(\mathbf{0})=O T(\mathbf{0})=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4331 | 451 | 1.000 | T\left(c_{1} \mathbf{v}_{1}+\cdots+\right. | | |
| johnston-linear-matrix-algebra_FO4332 | 451 | 1.000 | \left.c_{k} \mathbf{v}_{k}\right)=c_{1} T\left(\mathbf{v}_{1}\right)+\cdots+c_{k} T\left(\mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4333 | 451 | 1.000 | \mathbf{v}_{1}, \ldots, \mathbf{v}_{k} \in | | |
| johnston-linear-matrix-algebra_FO4334 | 451 | 1.000 | T\left(c_{1} \mathbf{v}_{1}\right)=c_{1} T\left(\mathbf{v}_{1}\right) | | |
| johnston-linear-matrix-algebra_FO4335 | 451 | 1.000 | c_{1}=c_{2}=1 | | |
| johnston-linear-matrix-algebra_FO4336 | 451 | 1.000 | T\left(\mathbf{v}_{1}+\mathbf{v}_{2}\right)=T\left(\mathbf{v}_{1}\right)+T\left(\mathbf{v}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO4337 | 451 | 1.000 | T\left(c_{1} \mathbf{v}_{1}+\cdots+c_{k} \mathbf{v}_{k}\right)=T\left(c_{1} \mathbf{v}_{1}\right)+T\left(c_{2} \mathbf{v}_{2}+\right. | | |
| johnston-linear-matrix-algebra_FO4338 | 451 | 1.000 | \left.\cdots+c_{k} \mathbf{v}_{k}\right)=c_{1} T\left(\mathbf{v}_{1}\right)+T\left(c_{2} \mathbf{v}_{2}+\cdots+c_{k} \mathbf{v}_{k}\right)=\cdots= | | |
| johnston-linear-matrix-algebra_FO4339 | 451 | 1.000 | c_{1} T\left(\mathbf{v}_{1}\right)+\cdots+c_{k} T\left(\mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4340 | 452 | 1.000 | [S+T] \mathbf{v}=(S+ | | |
| johnston-linear-matrix-algebra_FO4341 | 452 | 0.977 | T)(\mathbf{v})=S(\mathbf{v})+T(\mathbf{v})=[S] \mathbf{v}+[T] \mathbf{v}=([S]+ | | |
| johnston-linear-matrix-algebra_FO4342 | 452 | 1.000 | [T]) \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4343 | 452 | 1.000 | [S+T] \mathbf{v}=([S]+[T]) \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4344 | 452 | 1.000 | [c S] \mathbf{v}=(c S)(\mathbf{v})=c S(\mathbf{v})=c[S] \mathbf{v}=(c[S]) \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4345 | 452 | 1.000 | [c S] \mathbf{v}=(c[S]) \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4346 | 452 | 1.000 | [c S]=c[S] | | |
| johnston-linear-matrix-algebra_FO4347 | 452 | 1.000 | 160 \pi / 4=40 \pi | | |
| johnston-linear-matrix-algebra_FO4348 | 452 | 1.000 | 40 \pi | | |
| johnston-linear-matrix-algebra_FO4349 | 452 | 1.000 | A^{160}=I | | |
| johnston-linear-matrix-algebra_FO4350 | 452 | 0.974 | \cos (\theta) \sin (\theta)-\cos (\theta) \sin (\theta)=0 | | |
| johnston-linear-matrix-algebra_FO4351 | 452 | 1.000 | \mathbf{u}=(1, m) | | |
| johnston-linear-matrix-algebra_FO4352 | 452 | 1.000 | E_{i, j}=\mathbf{e}_{i} \mathbf{e}_{j}^{T} | | |
| johnston-linear-matrix-algebra_FO4353 | 452 | 1.000 | E_{i, j}^{2}= | | |
| johnston-linear-matrix-algebra_FO4354 | 452 | 0.978 | \mathbf{e}_{i} \mathbf{e}_{j}^{T} \mathbf{e}_{i} \mathbf{e}_{j}^{T}=\mathbf{e}_{i}\left(\mathbf{e}_{j} \cdot \mathbf{e}_{i}\right) \mathbf{e}_{j}^{T}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4355 | 452 | 1.000 | u_{1}^{2}=u_{3}^{2}=3 / 7 | | |
| johnston-linear-matrix-algebra_FO4356 | 452 | 1.000 | u_{2}^{2}=1 / 7 | | |
| johnston-linear-matrix-algebra_FO4357 | 452 | 1.000 | \cos (\theta)=1 / 8 | | |
| johnston-linear-matrix-algebra_FO4358 | 452 | 1.000 | (\sqrt{3}, 1, \sqrt{3}) / \sqrt{7} | | |
| johnston-linear-matrix-algebra_FO4359 | 452 | 1.000 | \theta=-\arccos (1 / 8) \approx | | |
| johnston-linear-matrix-algebra_FO4360 | 452 | 1.000 | \theta=\arccos (1 / 8) \approx 1.4455 | | |
| johnston-linear-matrix-algebra_FO4361 | 452 | 1.000 | \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO4362 | 453 | 1.000 | (A B)^{3}=(A B)(A B)(A B) | | |
| johnston-linear-matrix-algebra_FO4363 | 453 | 1.000 | A^{3} B^{3}=A A A B B B | | |
| johnston-linear-matrix-algebra_FO4364 | 453 | 1.000 | R(S(T(\mathbf{v}))) | | |
| johnston-linear-matrix-algebra_FO4365 | 453 | 0.774 | \mathbf{u}=(1,0) | | |
| johnston-linear-matrix-algebra_FO4366 | 453 | 1.000 | n \theta | | |
| johnston-linear-matrix-algebra_FO4367 | 453 | 1.000 | \|(2,1,0) \times(-2,3,0)\|=\|(0,0,8)\|=8 | | |
| johnston-linear-matrix-algebra_FO4368 | 453 | 1.000 | \|(1,2,3) \times(3,-1,2)\|=\|(7,7,-7)\|=7 \sqrt{3} | | |
| johnston-linear-matrix-algebra_FO4369 | 453 | 1.000 | \frac{1}{2}\|(0,4,0) \times(1,1,0)\|=\frac{1}{2}\|(0,0,-4)\|=2 | | |
| johnston-linear-matrix-algebra_FO4370 | 453 | 1.000 | \frac{1}{2}\|(-1,1,-1) \times(3,2,1)\|=\frac{1}{2}\|(3,-2,-5)\|= | | |
| johnston-linear-matrix-algebra_FO4371 | 453 | 1.000 | \frac{\sqrt{38}}{2} | | |
| johnston-linear-matrix-algebra_FO4372 | 453 | 0.826 | 1 \cdot 2 \cdot 3=6 | | |
| johnston-linear-matrix-algebra_FO4373 | 453 | 1.000 | |\mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})|=|(1,1,1) \cdot(3,4,-2)|=\mid 3+4- | | |
| johnston-linear-matrix-algebra_FO4374 | 453 | 0.958 | 2 \mid=5 | | |
| johnston-linear-matrix-algebra_FO4375 | 453 | 1.000 | \|(0,2,0) \times(-2,3,0)\|=\|(0,0,4)\|=4 | | |
| johnston-linear-matrix-algebra_FO4376 | 453 | 0.996 | \mid(1,0,0) | | |
| johnston-linear-matrix-algebra_FO4377 | 453 | 0.998 | ((-3,2,0) \times(2,-7,3)) \mid=6 | | |
| johnston-linear-matrix-algebra_FO4378 | 453 | 0.999 | \|\mathbf{v} \times \mathbf{w}\|=\|\mathbf{v}\|\|\mathbf{w}\| \sin (\theta) | | |
| johnston-linear-matrix-algebra_FO4379 | 453 | 1.000 | \|\mathbf{v}\|=\|\mathbf{w}\|=1 | | |
| johnston-linear-matrix-algebra_FO4380 | 453 | 1.000 | \|\mathbf{v} \times \mathbf{w}\|=\sqrt{(1 / 3)^{2}+(2 / 3)^{2}+(2 / 3)^{2}}=1 | | |
| johnston-linear-matrix-algebra_FO4381 | 453 | 1.000 | 1=\sin (\theta) | | |
| johnston-linear-matrix-algebra_FO4382 | 454 | 1.000 | \mathbf{v}=\mathbf{e}_{1}, \mathbf{w}=\mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO4383 | 454 | 1.000 | \mathbf{x}=\mathbf{e}_{2} | | |
| johnston-linear-matrix-algebra_FO4384 | 454 | 1.000 | (\mathbf{v} \times \mathbf{w}) \times \mathbf{x}= | | |
| johnston-linear-matrix-algebra_FO4385 | 454 | 1.000 | \left(\mathbf{e}_{1} \times \mathbf{e}_{2}\right) \times \mathbf{e}_{2}=\mathbf{e}_{3} \times \mathbf{e}_{2}=-\mathbf{e}_{1} | | |
| johnston-linear-matrix-algebra_FO4386 | 454 | 1.000 | \mathbf{v} \times(\mathbf{w} \times \mathbf{x})= | | |
| johnston-linear-matrix-algebra_FO4387 | 454 | 1.000 | \mathbf{e}_{1} \times\left(\mathbf{e}_{2} \times \mathbf{e}_{2}\right)=\mathbf{e}_{1} \times \mathbf{0}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4388 | 454 | 1.000 | \mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})=\mathbf{w} \cdot(\mathbf{x} \times \mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO4389 | 454 | 1.000 | \mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})= | | |
| johnston-linear-matrix-algebra_FO4390 | 454 | 1.000 | \mathbf{w} \cdot(\mathbf{x} \times \mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO4391 | 454 | 1.000 | \mathbf{w} \cdot(\mathbf{x} \times \mathbf{v})=\mathbf{x} \cdot(\mathbf{v} \times \mathbf{w}) | | |
| johnston-linear-matrix-algebra_FO4392 | 455 | 1.000 | \left[A^{2}\right]_{1,2}=1 | | |
| johnston-linear-matrix-algebra_FO4393 | 455 | 1.000 | \left[A^{4}\right]_{1,4}=9 | | |
| johnston-linear-matrix-algebra_FO4394 | 455 | 1.000 | 0 \leq k \leq n-1 | | |
| johnston-linear-matrix-algebra_FO4395 | 455 | 1.000 | n-k-1 | | |
| johnston-linear-matrix-algebra_FO4396 | 455 | 1.000 | k \geq 1,\left[A^{k}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO4397 | 455 | 0.983 | \left[A^{k}\right]_{i, 1} a_{1, j} | | |
| johnston-linear-matrix-algebra_FO4398 | 455 | 1.000 | \left[A^{k}\right]_{i, 1} | | |
| johnston-linear-matrix-algebra_FO4399 | 455 | 1.000 | k+1 | | |
| johnston-linear-matrix-algebra_FO4400 | 455 | 1.000 | \left[A^{k}\right]_{i, 2} a_{2, j} | | |
| johnston-linear-matrix-algebra_FO4401 | 455 | 1.000 | \left[A^{k+1}\right]_{i, j} | | |
| johnston-linear-matrix-algebra_FO4402 | 455 | 1.000 | \left[\begin{array}{ll}1 & 2 \\ 0 & 0 \\ 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4403 | 455 | 1.000 | \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4404 | 455 | 1.000 | \left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4405 | 455 | 1.000 | \left[\begin{array}{cccccc}1 & 0 & 0 & 0 & -4 / 3 & 2 / 5 \\ 0 & 1 & 0 & 0 & -4 / 3 & -4 / 5 \\ 0 & 0 & 1 & 0 & 7 / 3 & 9 / 20 \\ 0 & 0 & 0 & 1 & 2 & 1 / 10\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4406 | 455 | 1.000 | y: x | | |
| johnston-linear-matrix-algebra_FO4407 | 455 | 1.000 | (x, y)=(1,1) | | |
| johnston-linear-matrix-algebra_FO4408 | 456 | 1.000 | v, w | | |
| johnston-linear-matrix-algebra_FO4409 | 456 | 0.998 | v, w, x | | |
| johnston-linear-matrix-algebra_FO4410 | 456 | 1.000 | (v, w, x, y, z)= | | |
| johnston-linear-matrix-algebra_FO4411 | 456 | 0.999 | (-30,3,60,7,0)+z(-51,3,106,5,6) / 6 | | |
| johnston-linear-matrix-algebra_FO4412 | 456 | 0.978 | (x, y | | |
| johnston-linear-matrix-algebra_FO4413 | 456 | 0.976 | (\sin (1), \sqrt[3]{5}) | | |
| johnston-linear-matrix-algebra_FO4414 | 456 | 0.848 | (0,1,1) | | |
| johnston-linear-matrix-algebra_FO4415 | 456 | 1.000 | x=1, y=0 | | |
| johnston-linear-matrix-algebra_FO4416 | 456 | 1.000 | t \neq 1 / 2 | | |
| johnston-linear-matrix-algebra_FO4417 | 456 | 1.000 | t=1 / 2 | | |
| johnston-linear-matrix-algebra_FO4418 | 456 | 1.000 | (w, x, y, z)=(3,-24,30,0) | | |
| johnston-linear-matrix-algebra_FO4419 | 456 | 0.999 | (w, x, y, z)=(-4,60,-180,140) | | |
| johnston-linear-matrix-algebra_FO4420 | 456 | 1.000 | v_{2}=-2 | | |
| johnston-linear-matrix-algebra_FO4421 | 456 | 1.000 | v_{1}=1 | | |
| johnston-linear-matrix-algebra_FO4422 | 456 | 1.000 | \mathbf{v}=(1,-2,1) | | |
| johnston-linear-matrix-algebra_FO4423 | 456 | 1.000 | v_{4}=9 | | |
| johnston-linear-matrix-algebra_FO4424 | 456 | 1.000 | v_{3}=-5, v_{2}=-8 | | |
| johnston-linear-matrix-algebra_FO4425 | 456 | 1.000 | v_{1}=7 | | |
| johnston-linear-matrix-algebra_FO4426 | 456 | 1.000 | \mathbf{v}=(7,-8,-5,9) | | |
| johnston-linear-matrix-algebra_FO4427 | 456 | 1.000 | (x, y)=(1,2) | | |
| johnston-linear-matrix-algebra_FO4428 | 456 | 1.000 | \mathbf{b}=(1 / 2) \mathbf{v}_{1} | | |
| johnston-linear-matrix-algebra_FO4429 | 456 | 1.000 | \mathbf{b}=\mathbf{v}_{1}+5 \mathbf{v}_{2}-6 \mathbf{v}_{3} | | |
| johnston-linear-matrix-algebra_FO4430 | 456 | 1.000 | \mathbf{b}=\mathbf{v}_{1}+3 \mathbf{v}_{2} | | |
| johnston-linear-matrix-algebra_FO4431 | 457 | 1.000 | \mathbf{b}=-\mathbf{v}_{1}+\mathbf{v}_{2}-2 \mathbf{v}_{3} | | |
| johnston-linear-matrix-algebra_FO4432 | 457 | 1.000 | R_{1} | | |
| johnston-linear-matrix-algebra_FO4433 | 457 | 1.000 | w_{1} R_{1} | | |
| johnston-linear-matrix-algebra_FO4434 | 457 | 1.000 | R_{2} | | |
| johnston-linear-matrix-algebra_FO4435 | 457 | 1.000 | R_{1}-v_{1} R_{2} | | |
| johnston-linear-matrix-algebra_FO4436 | 457 | 1.000 | w_{1} \neq 0 | | |
| johnston-linear-matrix-algebra_FO4437 | 457 | 1.000 | \mathbf{w} \neq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4438 | 457 | 0.999 | x_{1}, x_{2}, x_{3} | | |
| johnston-linear-matrix-algebra_FO4439 | 457 | 1.000 | x_{3}=c\left(v_{1} w_{2}-\right. | | |
| johnston-linear-matrix-algebra_FO4440 | 457 | 0.865 | v_{2} w_{1} | | |
| johnston-linear-matrix-algebra_FO4441 | 457 | 0.865 | x_{1}=c\left(v_{2} w_{3}-v_{3} w_{2}\right) | | |
| johnston-linear-matrix-algebra_FO4442 | 457 | 1.000 | x_{2}=c\left(v_{3} w_{1}-v_{1} w_{3}\right) | | |
| johnston-linear-matrix-algebra_FO4443 | 457 | 0.950 | x=45 | | |
| johnston-linear-matrix-algebra_FO4444 | 457 | 0.950 | y=75 | | |
| johnston-linear-matrix-algebra_FO4445 | 457 | 1.000 | w=z | | |
| johnston-linear-matrix-algebra_FO4446 | 457 | 1.000 | x=3 z / 2 | | |
| johnston-linear-matrix-algebra_FO4447 | 457 | 1.000 | y=z | | |
| johnston-linear-matrix-algebra_FO4448 | 457 | 1.000 | v_{1}=18-2 v_{2}-2 v_{3} | | |
| johnston-linear-matrix-algebra_FO4449 | 457 | 1.000 | \mathbf{v}=\left(18-2 v_{2}-2 v_{3}, v_{2}, v_{3}\right) | | |
| johnston-linear-matrix-algebra_FO4450 | 457 | 1.000 | v_{2}, v_{3} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO4451 | 457 | 0.999 | 0=-1 / 3 | | |
| johnston-linear-matrix-algebra_FO4452 | 457 | 1.000 | T(1,0) | | |
| johnston-linear-matrix-algebra_FO4453 | 457 | 1.000 | T(0,1) | | |
| johnston-linear-matrix-algebra_FO4454 | 457 | 1.000 | c_{1}=c_{2}=c_{3}=1 / 2 | | |
| johnston-linear-matrix-algebra_FO4455 | 457 | 1.000 | c_{4}=-1 / 2 | | |
| johnston-linear-matrix-algebra_FO4456 | 457 | 1.000 | T(0,1)= | | |
| johnston-linear-matrix-algebra_FO4457 | 458 | 1.000 | u=1 / x, v=1 / y | | |
| johnston-linear-matrix-algebra_FO4458 | 458 | 1.000 | w=1 / z | | |
| johnston-linear-matrix-algebra_FO4459 | 458 | 1.000 | (u, v, w)=(2,-1,3) | | |
| johnston-linear-matrix-algebra_FO4460 | 458 | 1.000 | (x, y, z)=(1 / 2,-1,1 / 3) | | |
| johnston-linear-matrix-algebra_FO4461 | 458 | 0.995 | \left[\begin{array}{ll}a & b \\ c & d\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4462 | 458 | 1.000 | c=b, d=a | | |
| johnston-linear-matrix-algebra_FO4463 | 458 | 0.963 | \left[\begin{array}{cc}a & b \\ b & a\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4464 | 458 | 1.000 | c=b, d=a-b | | |
| johnston-linear-matrix-algebra_FO4465 | 458 | 0.998 | \left[\begin{array}{cc}a & b \\ b & a-b\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4466 | 458 | 1.000 | B C=C B | | |
| johnston-linear-matrix-algebra_FO4467 | 458 | 1.000 | B=\left[\begin{array}{cc}a & b \\ b & a-b\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4468 | 458 | 1.000 | a-b=a | | |
| johnston-linear-matrix-algebra_FO4469 | 458 | 0.689 | B=\left[\begin{array}{ll}a & 0 \\ 0 & a\end{array}\right]=a I | | |
| johnston-linear-matrix-algebra_FO4470 | 458 | 0.941 | a I | | |
| johnston-linear-matrix-algebra_FO4471 | 458 | 0.901 | (a I) A=a A=A(a I) | | |
| johnston-linear-matrix-algebra_FO4472 | 458 | 1.000 | (b, m)=(-1,3) | | |
| johnston-linear-matrix-algebra_FO4473 | 458 | 1.000 | y=3 x-1 | | |
| johnston-linear-matrix-algebra_FO4474 | 458 | 1.000 | (a, b, c)=(2,-3,4) | | |
| johnston-linear-matrix-algebra_FO4475 | 458 | 1.000 | y=2 x^{2}-3 x+4 | | |
| johnston-linear-matrix-algebra_FO4476 | 458 | 0.999 | d=4 | | |
| johnston-linear-matrix-algebra_FO4477 | 459 | 0.999 | \left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4478 | 459 | 1.000 | \left[\begin{array}{lll}1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4479 | 459 | 1.000 | \left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4480 | 459 | 1.000 | \left[\begin{array}{cc}-5 / 3 & 2 / 3 \\ 4 / 3 & -1 / 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4481 | 459 | 1.000 | \left[\begin{array}{ccc}0 & -1 / 3 & 2 / 3 \\ 0 & 1 / 3 & 1 / 3 \\ 1 & 1 / 3 & -5 / 3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4482 | 459 | 1.000 | \left[\begin{array}{ccc}0 & 0 & 1 \\ 0 & -1 / 2 & 3 / 2 \\ 1 & 1 / 2 & -3 / 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4483 | 459 | 1.000 | B=-I | | |
| johnston-linear-matrix-algebra_FO4484 | 459 | 1.000 | A+B=O | | |
| johnston-linear-matrix-algebra_FO4485 | 459 | 1.000 | A^{-1} 7 | | |
| johnston-linear-matrix-algebra_FO4486 | 459 | 1.000 | I=O | | |
| johnston-linear-matrix-algebra_FO4487 | 459 | 1.000 | X=B A^{-1} | | |
| johnston-linear-matrix-algebra_FO4488 | 459 | 1.000 | A^{-1} B | | |
| johnston-linear-matrix-algebra_FO4489 | 459 | 1.000 | A=(a-b) I_{n}+b J_{n} | | |
| johnston-linear-matrix-algebra_FO4490 | 459 | 1.000 | B=c I_{n}+d J_{n} | | |
| johnston-linear-matrix-algebra_FO4491 | 460 | 1.000 | c=1 /(a-b) | | |
| johnston-linear-matrix-algebra_FO4492 | 460 | 1.000 | b c+ | | |
| johnston-linear-matrix-algebra_FO4493 | 460 | 0.995 | d(a-b)+n b d=0 | | |
| johnston-linear-matrix-algebra_FO4494 | 460 | 0.995 | d=-b(n d+1 /(a- | | |
| johnston-linear-matrix-algebra_FO4495 | 460 | 0.992 | b)) /(a-b) | | |
| johnston-linear-matrix-algebra_FO4496 | 460 | 0.985 | A^{2}=\left[\begin{array}{cc}7 & 4 \\ 12 & 7\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4497 | 460 | 1.000 | A^{-2}=\left[\begin{array}{cc}7 & -4 \\ -12 & 7\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4498 | 460 | 1.000 | m \times n, A^{T} A | | |
| johnston-linear-matrix-algebra_FO4499 | 460 | 1.000 | \left(A^{T} A\right)^{-1} | | |
| johnston-linear-matrix-algebra_FO4500 | 460 | 1.000 | P^{T}=\left(A\left(A^{T} A\right)^{-1} A^{T}\right)^{T}=\left(A^{T}\right)^{T}\left(\left(A^{T} A\right)^{-1}\right)^{T} A^{T}= | | |
| johnston-linear-matrix-algebra_FO4501 | 460 | 0.968 | A\left(\left(A^{T} A\right)^{T}\right)^{-1} A^{T}=A\left(A^{T} A\right)^{-1} A^{T}=P | | |
| johnston-linear-matrix-algebra_FO4502 | 460 | 1.000 | E_{k} \cdots E_{2} E_{1} P=I | | |
| johnston-linear-matrix-algebra_FO4503 | 460 | 1.000 | E_{k} \cdots E_{2} E_{1}=P^{-1} | | |
| johnston-linear-matrix-algebra_FO4504 | 460 | 1.000 | P^{-1} Q | | |
| johnston-linear-matrix-algebra_FO4505 | 460 | 1.000 | X= | | |
| johnston-linear-matrix-algebra_FO4506 | 460 | 1.000 | X=A^{-1} I B^{-1}=A^{-1} B^{-1} | | |
| johnston-linear-matrix-algebra_FO4507 | 460 | 1.000 | a_{n, n} x_{n}=0 | | |
| johnston-linear-matrix-algebra_FO4508 | 460 | 1.000 | x_{n}=0 | | |
| johnston-linear-matrix-algebra_FO4509 | 460 | 1.000 | a_{n, n} \neq 0 | | |
| johnston-linear-matrix-algebra_FO4510 | 460 | 0.997 | a_{n-1, n-1} x_{n-1}=0 | | |
| johnston-linear-matrix-algebra_FO4511 | 460 | 0.997 | x_{n-1}=0 | | |
| johnston-linear-matrix-algebra_FO4512 | 460 | 1.000 | a_{n-1, n-1} \neq 0 | | |
| johnston-linear-matrix-algebra_FO4513 | 460 | 1.000 | a_{j, j}=0 | | |
| johnston-linear-matrix-algebra_FO4514 | 460 | 1.000 | a_{j, j} x_{j}=0 | | |
| johnston-linear-matrix-algebra_FO4515 | 460 | 1.000 | x_{j} | | |
| johnston-linear-matrix-algebra_FO4516 | 460 | 1.000 | A \mathbf{x}_{j}=\mathbf{e}_{j} | | |
| johnston-linear-matrix-algebra_FO4517 | 460 | 0.994 | n-j | | |
| johnston-linear-matrix-algebra_FO4518 | 460 | 1.000 | \mathbf{x}_{j} | | |
| johnston-linear-matrix-algebra_FO4519 | 460 | 0.999 | b_{1}, b_{2}, \ldots, b_{n} | | |
| johnston-linear-matrix-algebra_FO4520 | 460 | 1.000 | b_{1}=1 / a_{1}, b_{2}=1 / a_{2} | | |
| johnston-linear-matrix-algebra_FO4521 | 460 | 0.999 | \left(A^{-1}\right)^{-1} | | |
| johnston-linear-matrix-algebra_FO4522 | 460 | 1.000 | A^{-1}\left(A^{-1}\right)^{-1}=\left(A^{-1}\right)^{-1} A^{-1}=I | | |
| johnston-linear-matrix-algebra_FO4523 | 460 | 1.000 | \left(A^{-1}\right)^{T} A^{T}=A^{T}\left(A^{-1}\right)^{T}=I | | |
| johnston-linear-matrix-algebra_FO4524 | 460 | 0.997 | I=I^{T}=\left(A A^{-1}\right)^{T}=\left(A^{-1}\right)^{T} A^{T} | | |
| johnston-linear-matrix-algebra_FO4525 | 460 | 1.000 | I=I^{T}=\left(A^{-1} A\right)^{T}=A^{T}\left(A^{-1}\right)^{T} | | |
| johnston-linear-matrix-algebra_FO4526 | 460 | 1.000 | A^{-1} k | | |
| johnston-linear-matrix-algebra_FO4527 | 460 | 1.000 | A A^{-1} | | |
| johnston-linear-matrix-algebra_FO4528 | 460 | 1.000 | A^{k}\left(A^{-1}\right)^{k}=I | | |
| johnston-linear-matrix-algebra_FO4529 | 460 | 1.000 | \left(A^{-1}\right)^{k} A^{k}=I | | |
| johnston-linear-matrix-algebra_FO4530 | 461 | 1.000 | k, \ell \geq 0 | | |
| johnston-linear-matrix-algebra_FO4531 | 461 | 1.000 | k, \ell<0 | | |
| johnston-linear-matrix-algebra_FO4532 | 461 | 1.000 | \ell<0 | | |
| johnston-linear-matrix-algebra_FO4533 | 461 | 1.000 | A^{k} A^{\ell}=(A A \cdots A)\left(A^{-1} A^{-1} \cdots A^{-1}\right) | | |
| johnston-linear-matrix-algebra_FO4534 | 461 | 1.000 | |\ell|=-\ell | | |
| johnston-linear-matrix-algebra_FO4535 | 461 | 1.000 | k-|\ell|=k+\ell | | |
| johnston-linear-matrix-algebra_FO4536 | 461 | 1.000 | k \geq|\ell| | | |
| johnston-linear-matrix-algebra_FO4537 | 461 | 1.000 | |\ell|-k=-(k+\ell) | | |
| johnston-linear-matrix-algebra_FO4538 | 461 | 1.000 | |\ell|>k | | |
| johnston-linear-matrix-algebra_FO4539 | 461 | 1.000 | k \geq 0, \ell<0 | | |
| johnston-linear-matrix-algebra_FO4540 | 461 | 1.000 | \left(A^{k}\right)^{\ell}= | | |
| johnston-linear-matrix-algebra_FO4541 | 461 | 0.999 | (A A \cdots A)^{-1}(A A \cdots A)^{-1} \cdots(A A \cdots A)^{-1} | | |
| johnston-linear-matrix-algebra_FO4542 | 461 | 1.000 | (A A \cdots A)^{-1}= | | |
| johnston-linear-matrix-algebra_FO4543 | 461 | 1.000 | A^{-1} A^{-1} \cdots A^{-1} | | |
| johnston-linear-matrix-algebra_FO4544 | 461 | 1.000 | k|\ell|=-k \ell | | |
| johnston-linear-matrix-algebra_FO4545 | 461 | 1.000 | A^{-|k \ell|}=A^{k \ell} | | |
| johnston-linear-matrix-algebra_FO4546 | 461 | 1.000 | B^{T} A^{T}=(A B)^{T}=I^{T}=I | | |
| johnston-linear-matrix-algebra_FO4547 | 461 | 1.000 | \left(A^{T}\right)^{-1}=B^{T} | | |
| johnston-linear-matrix-algebra_FO4548 | 461 | 0.992 | A_{1} B_{1,1}, A_{2} B_{2,2}, \ldots, A_{n} B_{n, n} | | |
| johnston-linear-matrix-algebra_FO4549 | 461 | 1.000 | \left[\begin{array}{cc}A & I \\ I & O\end{array}\right]\left[\begin{array}{cc}O & I \\ I & -A\end{array}\right]=\left[\begin{array}{cc}I & A-A \\ O & I\end{array}\right]= | | |
| johnston-linear-matrix-algebra_FO4550 | 462 | 0.977 | (\pi, 0) | | |
| johnston-linear-matrix-algebra_FO4551 | 462 | 1.000 | \frac{1}{2}(\pi, 0)=(\pi / 2,0) | | |
| johnston-linear-matrix-algebra_FO4552 | 462 | 1.000 | \operatorname{span}\{(-2,1)\} | | |
| johnston-linear-matrix-algebra_FO4553 | 462 | 0.955 | (0,-1) | | |
| johnston-linear-matrix-algebra_FO4554 | 462 | 0.955 | (1,0)+(0,-1)= | | |
| johnston-linear-matrix-algebra_FO4555 | 462 | 1.000 | x-y=0 | | |
| johnston-linear-matrix-algebra_FO4556 | 462 | 1.000 | -3 x+y+z=0 | | |
| johnston-linear-matrix-algebra_FO4557 | 462 | 1.000 | y=-x | | |
| johnston-linear-matrix-algebra_FO4558 | 462 | 0.523 | z) | | |
| johnston-linear-matrix-algebra_FO4559 | 462 | 0.523 | x+y+2 z= | | |
| johnston-linear-matrix-algebra_FO4560 | 462 | 1.000 | (1,0), \mathbf{v}_{2}=(0,1) | | |
| johnston-linear-matrix-algebra_FO4561 | 462 | 1.000 | \mathbf{v}_{3}=(1,1) | | |
| johnston-linear-matrix-algebra_FO4562 | 462 | 1.000 | \mathbf{v}=\mathbf{w}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4563 | 463 | 1.000 | 3-3 k \neq 0 | | |
| johnston-linear-matrix-algebra_FO4564 | 463 | 1.000 | k \neq 1 | | |
| johnston-linear-matrix-algebra_FO4565 | 463 | 1.000 | y+z-3 x=0 | | |
| johnston-linear-matrix-algebra_FO4566 | 463 | 1.000 | \left(c_{0}, c_{1}\right)=(-3,2) | | |
| johnston-linear-matrix-algebra_FO4567 | 463 | 1.000 | p(x)=2 x-3 | | |
| johnston-linear-matrix-algebra_FO4568 | 463 | 1.000 | \left(c_{0}, c_{1}, c_{2}\right)=(2,-2,1) | | |
| johnston-linear-matrix-algebra_FO4569 | 463 | 1.000 | x^{2}-2 x+2 | | |
| johnston-linear-matrix-algebra_FO4570 | 463 | 1.000 | A \mathbf{0}=\mathbf{0} \neq \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO4571 | 463 | 1.000 | A_{1}=1 | | |
| johnston-linear-matrix-algebra_FO4572 | 463 | 1.000 | A_{1}^{-1}=1 | | |
| johnston-linear-matrix-algebra_FO4573 | 463 | 0.996 | (1,2, \ldots, n) | | |
| johnston-linear-matrix-algebra_FO4574 | 463 | 0.998 | (n+1, n+2, \ldots, 2 n) | | |
| johnston-linear-matrix-algebra_FO4575 | 463 | 1.000 | \mathbf{v}_{1}+\mathbf{v}_{2} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4576 | 463 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4577 | 463 | 1.000 | c_{1} \mathbf{v}_{1} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4578 | 463 | 1.000 | \mathbf{v}_{1} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4579 | 463 | 0.998 | c_{2} \mathbf{v}_{2} \in \mathcal{S}, \ldots, c_{k} \mathbf{v}_{k} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4580 | 463 | 0.998 | c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4581 | 463 | 1.000 | \left(c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}\right)+c_{3} \mathbf{v}_{3} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4582 | 463 | 0.999 | c_{1} \mathbf{v}_{1}+\cdots+c_{k} \mathbf{v}_{k} \in \mathcal{S} | | |
| johnston-linear-matrix-algebra_FO4583 | 463 | 1.000 | (A-I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4584 | 463 | 0.508 | \operatorname{null}(A-I) | | |
| johnston-linear-matrix-algebra_FO4585 | 463 | 1.000 | c_{1} \mathbf{v}+c_{2} \mathbf{w}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4586 | 463 | 1.000 | c_{1}=0 | | |
| johnston-linear-matrix-algebra_FO4587 | 463 | 1.000 | c_{2} \neq 0 | | |
| johnston-linear-matrix-algebra_FO4588 | 463 | 1.000 | c_{1} \neq 0 | | |
| johnston-linear-matrix-algebra_FO4589 | 463 | 1.000 | \mathbf{v}=\left(-c_{2} / c_{1}\right) \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4590 | 463 | 1.000 | 0 \mathbf{v}+1 \mathbf{w}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4591 | 463 | 1.000 | 1 \mathbf{v}+(-c) \mathbf{w}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4592 | 463 | 1.000 | S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO4593 | 463 | 1.000 | c_{i} \neq 0 | | |
| johnston-linear-matrix-algebra_FO4594 | 463 | 0.990 | \mathbf{v}_{i}=\frac{-c_{1}}{c_{i}} \mathbf{v}_{1}+\ldots+\frac{-c_{i-1}}{c_{i}} \mathbf{v}_{i-1}+\frac{-c_{i+1}}{c_{i}} \mathbf{v}_{i+1}+\ldots+\frac{-c_{k}}{c_{i}} \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO4595 | 463 | 1.000 | B=\left\{\mathbf{0}, \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO4596 | 463 | 1.000 | C=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}, \mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{m}\right\} | | |
| johnston-linear-matrix-algebra_FO4597 | 463 | 1.000 | \mathbf{x} \in \operatorname{range}(A B) | | |
| johnston-linear-matrix-algebra_FO4598 | 463 | 1.000 | \mathbf{y} \in \mathbb{R}^{p} | | |
| johnston-linear-matrix-algebra_FO4599 | 463 | 1.000 | \mathbf{x}=(A B) \mathbf{y}=A(B \mathbf{y}) | | |
| johnston-linear-matrix-algebra_FO4600 | 463 | 0.999 | \mathbf{x} \in \operatorname{range}(A) | | |
| johnston-linear-matrix-algebra_FO4601 | 463 | 1.000 | \mathbf{x} \in \operatorname{null}(B) | | |
| johnston-linear-matrix-algebra_FO4602 | 463 | 1.000 | B \mathbf{x}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4603 | 463 | 1.000 | A B \mathbf{x}=A \mathbf{0}= | | |
| johnston-linear-matrix-algebra_FO4604 | 463 | 0.998 | \mathbf{x} \in \operatorname{null}(A B) | | |
| johnston-linear-matrix-algebra_FO4605 | 463 | 0.998 | \operatorname{null}(B) \subseteq | | |
| johnston-linear-matrix-algebra_FO4606 | 464 | 0.997 | \operatorname{range}(A B)=\operatorname{span}((1,0)) | | |
| johnston-linear-matrix-algebra_FO4607 | 464 | 0.999 | \operatorname{range}(B)=\operatorname{span}((0,1)) | | |
| johnston-linear-matrix-algebra_FO4608 | 464 | 1.000 | \operatorname{null}(A)=\operatorname{span}((1,0)) | | |
| johnston-linear-matrix-algebra_FO4609 | 464 | 0.992 | \operatorname{null}(A B)=\operatorname{span}((0,1))) | | |
| johnston-linear-matrix-algebra_FO4610 | 464 | 1.000 | (x, y)=(1,3) | | |
| johnston-linear-matrix-algebra_FO4611 | 464 | 1.000 | B=\{(1,3)\} | | |
| johnston-linear-matrix-algebra_FO4612 | 464 | 1.000 | B=\{(1,1,-3),(1,0,-2)\} | | |
| johnston-linear-matrix-algebra_FO4613 | 464 | 1.000 | z=3 | | |
| johnston-linear-matrix-algebra_FO4614 | 464 | 1.000 | y=4 | | |
| johnston-linear-matrix-algebra_FO4615 | 464 | 1.000 | (x, y, z)=(-1,4,3) | | |
| johnston-linear-matrix-algebra_FO4616 | 464 | 1.000 | B=\{(-1,4,3)\} | | |
| johnston-linear-matrix-algebra_FO4617 | 464 | 0.999 | \{(1,2,3,4),(3,5,7,9)\} | | |
| johnston-linear-matrix-algebra_FO4618 | 464 | 1.000 | \{(1,1,4,5,1),(2,-7,4,2,2),(5,2,4,5,3)\} | | |
| johnston-linear-matrix-algebra_FO4619 | 464 | 0.997 | \{(2,3,1,1),(3,3,2,3),(2,1,2,3),(1,0,0,0)\} | | |
| johnston-linear-matrix-algebra_FO4620 | 464 | 1.000 | d_{1}, d_{2}, \ldots, d_{n} \in \mathbb{R} | | |
| johnston-linear-matrix-algebra_FO4621 | 464 | 1.000 | d_{1} c_{1}=\cdots=d_{n} c_{n}=0 | | |
| johnston-linear-matrix-algebra_FO4622 | 464 | 1.000 | \ldots, c_{n} | | |
| johnston-linear-matrix-algebra_FO4623 | 464 | 1.000 | d_{1}=\cdots=d_{n}=0 | | |
| johnston-linear-matrix-algebra_FO4624 | 464 | 1.000 | \mathbf{v}_{j}=\left(1 / c_{j}\right)\left(c_{j} \mathbf{v}_{j}\right) | | |
| johnston-linear-matrix-algebra_FO4625 | 464 | 0.999 | \{(2,4,3,4),(3,2,-1,-1),(1,0,0,0),(0,1,0,0)\} | | |
| johnston-linear-matrix-algebra_FO4626 | 464 | 0.988 | (A):\{(1,0),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO4627 | 464 | 1.000 | \operatorname{null}(A):\{ \}(\operatorname{null}(A)=\{\mathbf{0}\}) | | |
| johnston-linear-matrix-algebra_FO4628 | 464 | 0.894 | \operatorname{range}\left(A^{T}\right):\{(1,0),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO4629 | 464 | 0.994 | \operatorname{null}\left(A^{T}\right):\{ \}\left(\operatorname{null}\left(A^{T}\right)=\{\mathbf{0}\}\right) | | |
| johnston-linear-matrix-algebra_FO4630 | 464 | 0.813 | \operatorname{range}(A):\{(0,1,0)\} | | |
| johnston-linear-matrix-algebra_FO4631 | 464 | 0.999 | \operatorname{null}(A):\{(1,0,0),(0,-2,1)\} | | |
| johnston-linear-matrix-algebra_FO4632 | 464 | 0.966 | \left(A^{T}\right):\{(0,1,2)\} | | |
| johnston-linear-matrix-algebra_FO4633 | 464 | 1.000 | \operatorname{null}\left(A^{T}\right):\{(1,0,0),(0,0,1)\} | | |
| johnston-linear-matrix-algebra_FO4634 | 464 | 0.608 | \operatorname{range}(A):\{(1,0,0),(0,2,1)\} | | |
| johnston-linear-matrix-algebra_FO4635 | 464 | 1.000 | \operatorname{null}\left(A^{T}\right):\{(0,-1,2)\} | | |
| johnston-linear-matrix-algebra_FO4636 | 464 | 0.928 | (A):\{(1,0,0),(0,1,0),(0,0,1)\} | | |
| johnston-linear-matrix-algebra_FO4637 | 464 | 0.699 | \operatorname{null}(A):\{(-2,1,0,0,0),(-3,0,1,1,0)\} | | |
| johnston-linear-matrix-algebra_FO4638 | 464 | 0.865 | \left(A^{T}\right):\{(1,2,0,3,0),(0,0,1,-1,0) | | |
| johnston-linear-matrix-algebra_FO4639 | 464 | 0.706 | \operatorname{range}(A):\{(0,-1,-2),(-4,2,0)\} | | |
| johnston-linear-matrix-algebra_FO4640 | 464 | 0.725 | \operatorname{null}(A):\{(1,0,1,0,0),(3,1 / 2,0,1,0) | | |
| johnston-linear-matrix-algebra_FO4641 | 464 | 0.546 | \left(A^{T}\right):\{(1,0,-1,-3,-3 / 2) | | |
| johnston-linear-matrix-algebra_FO4642 | 464 | 1.000 | \operatorname{null}\left(A^{T}\right):\{(-1,-2,1)\} | | |
| johnston-linear-matrix-algebra_FO4643 | 465 | 0.999 | \operatorname{rank}(A)=4 | | |
| johnston-linear-matrix-algebra_FO4644 | 465 | 1.000 | \operatorname{nullity}(A)=4 | | |
| johnston-linear-matrix-algebra_FO4645 | 465 | 0.998 | (A):\{(0,-6,4,-1),(2,-1,-2,0) | | |
| johnston-linear-matrix-algebra_FO4646 | 465 | 1.000 | (-1,1,1,0),(-1,6,-2,1)\} | | |
| johnston-linear-matrix-algebra_FO4647 | 465 | 0.904 | \operatorname{null}(A):\{(2,-1,0,0,0,0,0,0) | | |
| johnston-linear-matrix-algebra_FO4648 | 465 | 0.826 | \operatorname{range}\left(A^{T}\right):\{(1,2,0,1,0,-1,1,0) | | |
| johnston-linear-matrix-algebra_FO4649 | 465 | 1.000 | \{(1,1),(2,2)\} | | |
| johnston-linear-matrix-algebra_FO4650 | 465 | 0.978 | \operatorname{rank}(A+B) \leq | | |
| johnston-linear-matrix-algebra_FO4651 | 465 | 0.855 | A=B=I_{n} | | |
| johnston-linear-matrix-algebra_FO4652 | 465 | 0.855 | (A+B)=\operatorname{rank}\left(2 I_{n}\right)=n | | |
| johnston-linear-matrix-algebra_FO4653 | 465 | 1.000 | \operatorname{rank}(A)+\operatorname{rank}(B)=n+n=2 n | | |
| johnston-linear-matrix-algebra_FO4654 | 465 | 0.992 | \operatorname{rank}(A B) \leq | | |
| johnston-linear-matrix-algebra_FO4655 | 465 | 1.000 | \min \{\operatorname{rank}(A), \operatorname{rank}(B)\} | | |
| johnston-linear-matrix-algebra_FO4656 | 465 | 0.983 | \operatorname{rank}\left(A_{2}\right)=2 | | |
| johnston-linear-matrix-algebra_FO4657 | 465 | 1.000 | k \geq 3 | | |
| johnston-linear-matrix-algebra_FO4658 | 465 | 1.000 | R_{2}-2 R_{1} | | |
| johnston-linear-matrix-algebra_FO4659 | 465 | 1.000 | x \neq 2 | | |
| johnston-linear-matrix-algebra_FO4660 | 465 | 0.586 | x \neq \pm 1 | | |
| johnston-linear-matrix-algebra_FO4661 | 465 | 1.000 | A^{-1}(A B)=B | | |
| johnston-linear-matrix-algebra_FO4662 | 465 | 1.000 | \operatorname{rank}(B)=\operatorname{rank}\left(A^{-1}(A B)\right) \leq | | |
| johnston-linear-matrix-algebra_FO4663 | 466 | 0.523 | \operatorname{span}\left(\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\right)=\operatorname{span}\left(\mathbf{v}_{1}, \ldots, \mathbf{v}_{i-1}, \mathbf{v}_{i+1}, \ldots, \mathbf{v}_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4664 | 466 | 1.000 | \left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{i-1}, \mathbf{v}_{i+1}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO4665 | 466 | 1.000 | \{(1,-1,2),(2,-2,4)\} | | |
| johnston-linear-matrix-algebra_FO4666 | 466 | 1.000 | \mathbf{v}=A \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO4667 | 466 | 1.000 | \mathbf{w}= | | |
| johnston-linear-matrix-algebra_FO4668 | 466 | 1.000 | A^{T} \mathbf{x} | | |
| johnston-linear-matrix-algebra_FO4669 | 466 | 0.998 | [B \mid \mathbf{0}] | | |
| johnston-linear-matrix-algebra_FO4670 | 466 | 1.000 | S \mathbf{x} \neq \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4671 | 466 | 1.000 | \operatorname{null}(A) \neq \operatorname{null}(B) | | |
| johnston-linear-matrix-algebra_FO4672 | 466 | 1.000 | \operatorname{range}\left(A^{T}\right) \neq \operatorname{range}\left(B^{T}\right) | | |
| johnston-linear-matrix-algebra_FO4673 | 466 | 1.000 | \operatorname{rank}(A)=\operatorname{rank}\left(A^{T}\right) | | |
| johnston-linear-matrix-algebra_FO4674 | 466 | 1.000 | \operatorname{rank}(\mathbf{v})=\operatorname{rank}\left(\mathbf{w}^{T}\right)=1 | | |
| johnston-linear-matrix-algebra_FO4675 | 466 | 0.997 | \operatorname{rank}(A)=\operatorname{rank}\left(\mathbf{v w}^{T}\right) \leq 1 | | |
| johnston-linear-matrix-algebra_FO4676 | 466 | 1.000 | \operatorname{rank}(A) \neq 0 | | |
| johnston-linear-matrix-algebra_FO4677 | 466 | 0.748 | w_{2}, w_{3}, \ldots, w_{n} | | |
| johnston-linear-matrix-algebra_FO4678 | 466 | 1.000 | w_{2} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4679 | 466 | 1.000 | w_{3} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4680 | 466 | 1.000 | \mathbf{w}^{T}=\left(1, w_{2}, w_{3}, \ldots, w_{n}\right) | | |
| johnston-linear-matrix-algebra_FO4681 | 467 | 0.999 | [R \mid \mathbf{c}] | | |
| johnston-linear-matrix-algebra_FO4682 | 467 | 0.974 | R \mathbf{x}=\mathbf{c} | | |
| johnston-linear-matrix-algebra_FO4683 | 467 | 1.000 | \mathbf{c} \neq \mathbf{b} | | |
| johnston-linear-matrix-algebra_FO4684 | 467 | 0.637 | \operatorname{range}(A)=\{\mathbf{0}\} | | |
| johnston-linear-matrix-algebra_FO4685 | 467 | 1.000 | R=E_{k} \cdots E_{2} E_{1} A | | |
| johnston-linear-matrix-algebra_FO4686 | 467 | 1.000 | E_{1}^{-1} E_{2}^{-1} \cdots E_{k}^{-1} R | | |
| johnston-linear-matrix-algebra_FO4687 | 467 | 1.000 | P=E_{1}^{-1} E_{2}^{-1} \cdots E_{k}^{-1} | | |
| johnston-linear-matrix-algebra_FO4688 | 467 | 1.000 | P_{1}, P_{2} \in \mathcal{M}_{m} | | |
| johnston-linear-matrix-algebra_FO4689 | 467 | 1.000 | A=P_{1} R | | |
| johnston-linear-matrix-algebra_FO4690 | 467 | 1.000 | B=P_{2} R | | |
| johnston-linear-matrix-algebra_FO4691 | 467 | 1.000 | P_{1} P_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO4692 | 467 | 1.000 | P=P_{1} P_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO4693 | 467 | 1.000 | P=E_{1} E_{2} \cdots E_{k} | | |
| johnston-linear-matrix-algebra_FO4694 | 467 | 1.000 | A=E_{1} E_{2} \cdots E_{k} B | | |
| johnston-linear-matrix-algebra_FO4695 | 467 | 1.000 | (x, y, z)=(0,1,0) | | |
| johnston-linear-matrix-algebra_FO4696 | 467 | 0.999 | (w, x, y, z)=(0,1,0,0) | | |
| johnston-linear-matrix-algebra_FO4697 | 467 | 0.998 | (v, w, x, y, z)=(1,1,0,0,0) | | |
| johnston-linear-matrix-algebra_FO4698 | 467 | 1.000 | (x, y, z)=(1,0,1) | | |
| johnston-linear-matrix-algebra_FO4699 | 468 | 1.000 | (v, w, x, y, z)=(0,1,1,0,0) | | |
| johnston-linear-matrix-algebra_FO4700 | 468 | 0.982 | (v, w, x, y, z)=(4,-1,3,2,8) | | |
| johnston-linear-matrix-algebra_FO4701 | 468 | 1.000 | \mathbf{v}_{\mathrm{e}}-\mathbf{v}_{\mathrm{s}}= | | |
| johnston-linear-matrix-algebra_FO4702 | 468 | 1.000 | \mathbf{x}=(1,0,0,1,0,1,1,0,1) | | |
| johnston-linear-matrix-algebra_FO4703 | 468 | 1.000 | m_{j} | | |
| johnston-linear-matrix-algebra_FO4704 | 468 | 1.000 | B-I | | |
| johnston-linear-matrix-algebra_FO4705 | 468 | 1.000 | \left(x_{1}, x_{2}\right)=(3,0) | | |
| johnston-linear-matrix-algebra_FO4706 | 468 | 1.000 | \left(x_{1}, x_{2}\right)=(21,1) / 8 | | |
| johnston-linear-matrix-algebra_FO4707 | 468 | 1.000 | \left(x_{1}, x_{2}, x_{3}\right)=(3,11,23) / 18 | | |
| johnston-linear-matrix-algebra_FO4708 | 468 | 1.000 | \left(x_{1}, x_{2}, x_{3}, x_{4}\right)=(6,2,0,23) / 30 | | |
| johnston-linear-matrix-algebra_FO4709 | 468 | 1.000 | \left(x_{1}, x_{2}\right)=(3,1) / 2 | | |
| johnston-linear-matrix-algebra_FO4710 | 468 | 1.000 | \left(x_{1}, x_{2}, x_{3}\right)=(1,0,1) | | |
| johnston-linear-matrix-algebra_FO4711 | 468 | 0.995 | 7 / 4 | | |
| johnston-linear-matrix-algebra_FO4712 | 468 | 0.882 | \left(x_{1}, x_{2}, x_{3}, x_{4}\right)=(8,2,0,1) / 8 | | |
| johnston-linear-matrix-algebra_FO4713 | 468 | 1.000 | \mathbf{x}=(1,0), \mathbf{b}=(2,2) | | |
| johnston-linear-matrix-algebra_FO4714 | 469 | 1.000 | x_{j} \geq y_{j} | | |
| johnston-linear-matrix-algebra_FO4715 | 469 | 1.000 | c_{j} x_{j} \geq c_{j} y_{j} | | |
| johnston-linear-matrix-algebra_FO4716 | 469 | 1.000 | \left(x_{1}, x_{2}\right)=(0.6,0.8) | | |
| johnston-linear-matrix-algebra_FO4717 | 469 | 1.000 | x_{2} \geq 1 / 2 | | |
| johnston-linear-matrix-algebra_FO4718 | 469 | 0.999 | x_{2} \leq 4 / 5 | | |
| johnston-linear-matrix-algebra_FO4719 | 469 | 1.000 | 2 x_{2} \leq c | | |
| johnston-linear-matrix-algebra_FO4720 | 469 | 1.000 | x_{2} \leq c / 2 | | |
| johnston-linear-matrix-algebra_FO4721 | 469 | 1.000 | c / 2 | | |
| johnston-linear-matrix-algebra_FO4722 | 469 | 0.381 | 1 / 2, c / 2 | | |
| johnston-linear-matrix-algebra_FO4723 | 469 | 1.000 | x_{1} \leq 1-x_{2} / c | | |
| johnston-linear-matrix-algebra_FO4724 | 469 | 1.000 | x_{1} \geq | | |
| johnston-linear-matrix-algebra_FO4725 | 469 | 1.000 | x_{2} / c | | |
| johnston-linear-matrix-algebra_FO4726 | 469 | 1.000 | 0 \leq x_{1} \leq x_{2} | | |
| johnston-linear-matrix-algebra_FO4727 | 469 | 1.000 | A^{T} \mathbf{1}=\mathbf{1} | | |
| johnston-linear-matrix-algebra_FO4728 | 469 | 1.000 | \mathbf{1}=(1,1, \ldots, 1) | | |
| johnston-linear-matrix-algebra_FO4729 | 469 | 1.000 | \mathbf{y}=\mathbf{1} | | |
| johnston-linear-matrix-algebra_FO4730 | 469 | 1.000 | \left[\begin{array}{c}1 \\ -1\end{array}\right]\left[\begin{array}{ll}1 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4731 | 469 | 0.924 | \left[\begin{array}{ccc}2 & 4 & 0 \\ 1 & -2 & 0 \\ 2 & 0 & -1\end{array}\right]\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4732 | 469 | 1.000 | \left[\begin{array}{l}3 \\ 1\end{array}\right]\left[\begin{array}{ll}2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4733 | 469 | 1.000 | \left[\begin{array}{cc}2 & 6 \\ 5 & 4 \\ 3 & -2\end{array}\right]\left[\begin{array}{cccc}1 & 0 & 2 & 5 / 11 \\ 0 & 1 & -1 / 2 & -7 / 22\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4734 | 470 | 0.994 | \tilde{C} \in \mathcal{M}_{m, r} | | |
| johnston-linear-matrix-algebra_FO4735 | 470 | 0.994 | \tilde{R} \in \mathcal{M}_{r, n} | | |
| johnston-linear-matrix-algebra_FO4736 | 470 | 1.000 | \tilde{C} \tilde{R}=C P^{-1} P R=C R=A | | |
| johnston-linear-matrix-algebra_FO4737 | 470 | 1.000 | P R | | |
| johnston-linear-matrix-algebra_FO4738 | 470 | 1.000 | P_{1}, P_{2}, Q_{1} | | |
| johnston-linear-matrix-algebra_FO4739 | 470 | 1.000 | Q_{2} | | |
| johnston-linear-matrix-algebra_FO4740 | 470 | 1.000 | P_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO4741 | 470 | 1.000 | Q_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO4742 | 470 | 1.000 | Q=Q_{2}^{-1} Q_{1} | | |
| johnston-linear-matrix-algebra_FO4743 | 470 | 0.951 | L=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], U=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4744 | 470 | 0.994 | L=\left[\begin{array}{cc}1 & 0 \\ -1 & 1\end{array}\right], U=\left[\begin{array}{ccc}3 & 1 & 2 \\ 0 & -2 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4745 | 470 | 1.000 | L=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\ 1 & 3 & 1\end{array}\right], U=\left[\begin{array}{cc}1 & 2 \\ 0 & -1 \\ 0 & 0\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4746 | 470 | 1.000 | L=\left[\begin{array}{ccc}1 & 0 & 0 \\ 3 & 1 & 0 \\ -2 & -1 & 1\end{array}\right], U=\left[\begin{array}{ccc}1 & -4 & 5 \\ 0 & 3 & -7 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4747 | 470 | 0.995 | P=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right], L=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], U=\left[\begin{array}{ll}1 & 3 \\ 0 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4748 | 470 | 0.999 | P=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right], L=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right], U= | | |
| johnston-linear-matrix-algebra_FO4749 | 470 | 0.760 | \left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4750 | 470 | 1.000 | (x, y)=(5,-3) | | |
| johnston-linear-matrix-algebra_FO4751 | 470 | 1.000 | (x, y, z)=(-3,6,-2) | | |
| johnston-linear-matrix-algebra_FO4752 | 470 | 1.000 | A=[B \mid C] | | |
| johnston-linear-matrix-algebra_FO4753 | 470 | 1.000 | \widetilde{U}=\left[U \mid L^{-1} C\right] | | |
| johnston-linear-matrix-algebra_FO4754 | 471 | 1.000 | A=L_{1} U_{1}=L_{2} U_{2} | | |
| johnston-linear-matrix-algebra_FO4755 | 471 | 1.000 | L_{1}, L_{2}, U_{1} | | |
| johnston-linear-matrix-algebra_FO4756 | 471 | 1.000 | U_{2} | | |
| johnston-linear-matrix-algebra_FO4757 | 471 | 1.000 | L_{1}^{-1} L_{2}=U_{1} U_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO4758 | 471 | 1.000 | L_{1}^{-1} L_{2} | | |
| johnston-linear-matrix-algebra_FO4759 | 471 | 1.000 | U_{1} U_{2}^{-1} | | |
| johnston-linear-matrix-algebra_FO4760 | 471 | 1.000 | L_{1}^{-1} | | |
| johnston-linear-matrix-algebra_FO4761 | 471 | 1.000 | L_{2} | | |
| johnston-linear-matrix-algebra_FO4762 | 471 | 1.000 | L_{1}^{-1} L_{2}=I | | |
| johnston-linear-matrix-algebra_FO4763 | 471 | 1.000 | L_{1}=L_{2} | | |
| johnston-linear-matrix-algebra_FO4764 | 471 | 1.000 | L_{1} U_{1}=L_{2} U_{2} | | |
| johnston-linear-matrix-algebra_FO4765 | 471 | 1.000 | U_{1}=U_{2} | | |
| johnston-linear-matrix-algebra_FO4766 | 471 | 1.000 | \underset{\sim}{k} \times k | | |
| johnston-linear-matrix-algebra_FO4767 | 471 | 1.000 | \widetilde{L} \widetilde{U} | | |
| johnston-linear-matrix-algebra_FO4768 | 471 | 1.000 | \widetilde{L} | | |
| johnston-linear-matrix-algebra_FO4769 | 471 | 1.000 | \widetilde{U} | | |
| johnston-linear-matrix-algebra_FO4770 | 471 | 1.000 | \frac{1}{275}\left[\begin{array}{cccccc}171 & 67 & 16 & -3 & -28 & -14 \\ 67 & 134 & 32 & -6 & -56 & -28 \\ 17 & 34 & 82 & 19 & -6 & -3 \\ 1 & 2 & 21 & 82 & 32 & 16 \\ -13 & -26 & 2 & 34 & 134 & 67 \\ -40 & -80 & -15 & 20 & 95 & 185\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4771 | 471 | 1.000 | \widetilde{U}=D U | | |
| johnston-linear-matrix-algebra_FO4772 | 471 | 1.000 | A=L \widetilde{U} | | |
| johnston-linear-matrix-algebra_FO4773 | 471 | 1.000 | \widetilde{U}=D^{-1} U | | |
| johnston-linear-matrix-algebra_FO4774 | 471 | 1.000 | A=L D \widetilde{U} | | |
| johnston-linear-matrix-algebra_FO4775 | 471 | 1.000 | L D \widetilde{U}=L D D^{-1} U=L U=A | | |
| johnston-linear-matrix-algebra_FO4776 | 471 | 1.000 | L=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], D=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], U=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4777 | 471 | 1.000 | R_{1}+c R_{2} | | |
| johnston-linear-matrix-algebra_FO4778 | 472 | 1.000 | u_{1,1}=0 | | |
| johnston-linear-matrix-algebra_FO4779 | 472 | 1.000 | [\mathbf{v}]_{B}=(2,1 / 2) | | |
| johnston-linear-matrix-algebra_FO4780 | 472 | 1.000 | 2(3,0)+\frac{1}{2}(0,2)=(6,1) | | |
| johnston-linear-matrix-algebra_FO4781 | 472 | 1.000 | (6,1)=c_{1}(3,0)+c_{2}(0,2) | | |
| johnston-linear-matrix-algebra_FO4782 | 472 | 1.000 | [\mathbf{v}]_{B}=(1 / 2,3 / 2) | | |
| johnston-linear-matrix-algebra_FO4783 | 472 | 1.000 | (2,3,1)= | | |
| johnston-linear-matrix-algebra_FO4784 | 472 | 1.000 | \frac{1}{2}(1,0,-1)+\frac{3}{2}(1,2,1) | | |
| johnston-linear-matrix-algebra_FO4785 | 472 | 1.000 | B=\{(2,3),(2,-1)\} | | |
| johnston-linear-matrix-algebra_FO4786 | 472 | 1.000 | \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4787 | 472 | 0.999 | \left[\begin{array}{cc}11 / 10 & 5 / 2 \\ 3 / 10 & 1 / 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4788 | 472 | 1.000 | \left[\begin{array}{cc}2 & 0 \\ -5 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4789 | 472 | 1.000 | \left[\begin{array}{cc}2 & 0 \\ 0 & -3\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4790 | 472 | 1.000 | \left[\begin{array}{ccc}10 & -5 & 0 \\ 5 & -6 & -4 \\ -11 & 5 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4791 | 472 | 1.000 | \operatorname{tr}(A)=\operatorname{tr}(B)=2 | | |
| johnston-linear-matrix-algebra_FO4792 | 472 | 1.000 | \operatorname{tr}(A)=6 | | |
| johnston-linear-matrix-algebra_FO4793 | 472 | 1.000 | \operatorname{tr}(B)=3 | | |
| johnston-linear-matrix-algebra_FO4794 | 472 | 0.995 | n! | | |
| johnston-linear-matrix-algebra_FO4795 | 472 | 0.999 | P^{T} P | | |
| johnston-linear-matrix-algebra_FO4796 | 472 | 0.549 | \mathbf{p}_{i} \cdot \mathbf{p}_{j}=1 | | |
| johnston-linear-matrix-algebra_FO4797 | 472 | 1.000 | \mathbf{p}_{i} | | |
| johnston-linear-matrix-algebra_FO4798 | 472 | 1.000 | \mathbf{p}_{i} \cdot \mathbf{p}_{j}=0 | | |
| johnston-linear-matrix-algebra_FO4799 | 472 | 0.953 | P^{T} P=I | | |
| johnston-linear-matrix-algebra_FO4800 | 472 | 0.953 | P^{T}=P^{-1} | | |
| johnston-linear-matrix-algebra_FO4801 | 472 | 1.000 | p_{1,2} | | |
| johnston-linear-matrix-algebra_FO4802 | 472 | 1.000 | p_{2,2} | | |
| johnston-linear-matrix-algebra_FO4803 | 472 | 1.000 | p_{1,1}=-p_{1,2} / 2 | | |
| johnston-linear-matrix-algebra_FO4804 | 472 | 1.000 | p_{2,1}=-p_{2,2} | | |
| johnston-linear-matrix-algebra_FO4805 | 472 | 1.000 | P_{B \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO4806 | 472 | 1.000 | P I P^{-1}= | | |
| johnston-linear-matrix-algebra_FO4807 | 472 | 1.000 | P P^{-1}=I | | |
| johnston-linear-matrix-algebra_FO4808 | 472 | 1.000 | P A P^{-1}=A | | |
| johnston-linear-matrix-algebra_FO4809 | 472 | 1.000 | A=B=I \in \mathcal{M}_{2} | | |
| johnston-linear-matrix-algebra_FO4810 | 472 | 1.000 | \operatorname{tr}(A B)=2 | | |
| johnston-linear-matrix-algebra_FO4811 | 472 | 1.000 | \operatorname{tr}(A) \operatorname{tr}(B)=4 | | |
| johnston-linear-matrix-algebra_FO4812 | 472 | 1.000 | \operatorname{tr}(A)=\operatorname{tr}(B)= | | |
| johnston-linear-matrix-algebra_FO4813 | 472 | 1.000 | P \in \mathcal{M}_{2} | | |
| johnston-linear-matrix-algebra_FO4814 | 473 | 1.000 | d_{j} | | |
| johnston-linear-matrix-algebra_FO4815 | 473 | 1.000 | [\mathbf{w}]_{B}:[\mathbf{v}]_{B}= | | |
| johnston-linear-matrix-algebra_FO4816 | 473 | 0.999 | \left(c_{1}, c_{2}, \ldots, c_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4817 | 473 | 0.999 | [\mathbf{w}]_{B}=\left(d_{1}, d_{2}, \ldots, d_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4818 | 473 | 1.000 | [\mathbf{v}+\mathbf{w}]_{B}=\left(c_{1}+d_{1}, c_{2}+\right. | | |
| johnston-linear-matrix-algebra_FO4819 | 473 | 0.566 | \left.d_{2}, \ldots, c_{k}+d_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4820 | 473 | 0.566 | [\mathbf{v}]_{B}+ | | |
| johnston-linear-matrix-algebra_FO4821 | 473 | 1.000 | [\mathbf{v}]_{B}=\left(d_{1}, d_{2}, \ldots, d_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4822 | 473 | 0.956 | [c \mathbf{v}]_{B}=\left(c d_{1}, c d_{2}, \ldots, c d_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4823 | 473 | 1.000 | c[\mathbf{v}]_{B} | | |
| johnston-linear-matrix-algebra_FO4824 | 473 | 1.000 | [\mathbf{v}]_{B}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4825 | 473 | 1.000 | c_{1}=\cdots=c_{m}=0 | | |
| johnston-linear-matrix-algebra_FO4826 | 473 | 0.997 | [\mathbf{v}]_{B} \in \mathbb{R}^{k} | | |
| johnston-linear-matrix-algebra_FO4827 | 473 | 1.000 | \mathcal{S}: \mathbf{x}=[\mathbf{v}]_{B} | | |
| johnston-linear-matrix-algebra_FO4828 | 473 | 0.998 | C=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\} | | |
| johnston-linear-matrix-algebra_FO4829 | 473 | 1.000 | P_{C \leftarrow E} P_{E \leftarrow B}=P_{C \leftarrow B} | | |
| johnston-linear-matrix-algebra_FO4830 | 473 | 1.000 | [T]_{B}[\mathbf{v}]_{B}=[T(\mathbf{v})]_{B} | | |
| johnston-linear-matrix-algebra_FO4831 | 473 | 1.000 | \left(c_{1}, \ldots, c_{n}\right) | | |
| johnston-linear-matrix-algebra_FO4832 | 473 | 1.000 | [T(\mathbf{v})]_{B}=A[\mathbf{v}]_{B} | | |
| johnston-linear-matrix-algebra_FO4833 | 473 | 1.000 | [T]_{B}[\mathbf{v}]_{B}=A[\mathbf{v}]_{B} | | |
| johnston-linear-matrix-algebra_FO4834 | 473 | 1.000 | A=[T] B | | |
| johnston-linear-matrix-algebra_FO4835 | 473 | 1.000 | \operatorname{nullity}(A)=n-\operatorname{rank}(A) | | |
| johnston-linear-matrix-algebra_FO4836 | 474 | 1.000 | \operatorname{tr}(A+B) | | |
| johnston-linear-matrix-algebra_FO4837 | 474 | 1.000 | \operatorname{tr}(c A) | | |
| johnston-linear-matrix-algebra_FO4838 | 474 | 1.000 | \mathbf{e}_{i} \mathbf{e}_{j}^{T} | | |
| johnston-linear-matrix-algebra_FO4839 | 474 | 1.000 | C=\mathbf{e}_{j} \mathbf{e}_{k}^{T} | | |
| johnston-linear-matrix-algebra_FO4840 | 474 | 1.000 | i, j, k | | |
| johnston-linear-matrix-algebra_FO4841 | 474 | 1.000 | i \neq k | | |
| johnston-linear-matrix-algebra_FO4842 | 474 | 1.000 | f(A C B)=f(A B C) | | |
| johnston-linear-matrix-algebra_FO4843 | 474 | 1.000 | f\left(A \mathbf{e}_{i} \mathbf{e}_{k}^{T}\right)= | | |
| johnston-linear-matrix-algebra_FO4844 | 474 | 0.999 | A=\mathbf{e}_{j} \mathbf{e}_{i}^{T} | | |
| johnston-linear-matrix-algebra_FO4845 | 474 | 1.000 | j, k | | |
| johnston-linear-matrix-algebra_FO4846 | 474 | 1.000 | A=\sum_{j, k=1}^{n} a_{j, k} \mathbf{e}_{j} \mathbf{e}_{k}^{T} | | |
| johnston-linear-matrix-algebra_FO4847 | 474 | 0.999 | B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\},[\mathbf{v}]_{B}=\left(c_{1}, c_{2}, \ldots, c_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4848 | 474 | 1.000 | \mathbf{v}_{i} \cdot \mathbf{v}_{j}=1 | | |
| johnston-linear-matrix-algebra_FO4849 | 474 | 1.000 | \mathbf{v}_{i} \cdot \mathbf{v}_{j}=0 | | |
| johnston-linear-matrix-algebra_FO4850 | 474 | 1.000 | B=\{(1,1),(0,1)\} | | |
| johnston-linear-matrix-algebra_FO4851 | 474 | 1.000 | (1,0), \mathbf{w}=(0,1) | | |
| johnston-linear-matrix-algebra_FO4852 | 474 | 1.000 | [\mathbf{v}]_{B}=(1,-1) | | |
| johnston-linear-matrix-algebra_FO4853 | 474 | 1.000 | \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k} \in B | | |
| johnston-linear-matrix-algebra_FO4854 | 474 | 1.000 | \mathbf{v}_{1} \cdot \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4855 | 474 | 1.000 | \left\|\mathbf{v}_{1}\right\| \neq 0 | | |
| johnston-linear-matrix-algebra_FO4856 | 474 | 1.000 | \mathbf{v}_{2} \cdot \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4857 | 474 | 1.000 | c_{2}=0 | | |
| johnston-linear-matrix-algebra_FO4858 | 474 | 1.000 | \mathbf{v}_{k} \cdot \mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4859 | 474 | 1.000 | c_{k}=0 | | |
| johnston-linear-matrix-algebra_FO4860 | 474 | 0.976 | \sqrt{\sin ^{2}(\theta)+\cos ^{2}(\theta)}=\sqrt{1}= | | |
| johnston-linear-matrix-algebra_FO4861 | 474 | 1.000 | \left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\} | | |
| johnston-linear-matrix-algebra_FO4862 | 474 | 1.000 | \mathbf{u}_{1}=(a, b) | | |
| johnston-linear-matrix-algebra_FO4863 | 474 | 1.000 | |a| \leq 1 | | |
| johnston-linear-matrix-algebra_FO4864 | 474 | 0.999 | \cos (\theta)=a | | |
| johnston-linear-matrix-algebra_FO4865 | 474 | 0.999 | b= \pm \sqrt{1-\cos ^{2}(\theta)}= | | |
| johnston-linear-matrix-algebra_FO4866 | 474 | 1.000 | \pm|\sin (\theta)| | | |
| johnston-linear-matrix-algebra_FO4867 | 474 | 1.000 | \mathbf{u}_{1}=(\cos (\theta), \sin (\theta)) | | |
| johnston-linear-matrix-algebra_FO4868 | 474 | 1.000 | -\theta | | |
| johnston-linear-matrix-algebra_FO4869 | 474 | 1.000 | \mathbf{u}_{1}= | | |
| johnston-linear-matrix-algebra_FO4870 | 475 | 1.000 | \mathbf{u}_{2} | | |
| johnston-linear-matrix-algebra_FO4871 | 475 | 1.000 | \mathbf{u}_{1} | | |
| johnston-linear-matrix-algebra_FO4872 | 475 | 0.953 | (-\sin (\theta), \cos (\theta)) | | |
| johnston-linear-matrix-algebra_FO4873 | 475 | 0.817 | \mathbf{u}_{2}= | | |
| johnston-linear-matrix-algebra_FO4874 | 475 | 1.000 | 2^{7} 3^{3}=3456 | | |
| johnston-linear-matrix-algebra_FO4875 | 475 | 1.000 | \operatorname{det}(-A)=(-1)^{n} \operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO4876 | 475 | 1.000 | A=I_{2} | | |
| johnston-linear-matrix-algebra_FO4877 | 475 | 1.000 | B=-I_{2} | | |
| johnston-linear-matrix-algebra_FO4878 | 475 | 1.000 | \operatorname{det}(A+B)=\operatorname{det}(O)=0 | | |
| johnston-linear-matrix-algebra_FO4879 | 475 | 1.000 | \operatorname{det}(A)+ | | |
| johnston-linear-matrix-algebra_FO4880 | 475 | 1.000 | \operatorname{det}(B)=1+1=2 | | |
| johnston-linear-matrix-algebra_FO4881 | 475 | 0.980 | \operatorname{det}\left(\left[P_{\mathbf{u}}\right]\right)=0 | | |
| johnston-linear-matrix-algebra_FO4882 | 475 | 1.000 | \mathbf{u}_{2}=(\sin (\theta),-\cos (\theta)) | | |
| johnston-linear-matrix-algebra_FO4883 | 475 | 1.000 | \mathbf{u}_{1}=R^{\theta}\left(\mathbf{e}_{1}\right) | | |
| johnston-linear-matrix-algebra_FO4884 | 475 | 1.000 | \mathbf{u}_{2}=R^{\theta}\left(\mathbf{e}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO4885 | 475 | 1.000 | \sin (\theta)=\cos (\theta-\pi / 2) | | |
| johnston-linear-matrix-algebra_FO4886 | 475 | 0.650 | -\cos (\theta)=\sin (\theta-\pi / 2) | | |
| johnston-linear-matrix-algebra_FO4887 | 475 | 1.000 | R_{\theta-\pi / 2}\left(\mathbf{e}_{2}\right) | | |
| johnston-linear-matrix-algebra_FO4888 | 475 | 1.000 | \mathbf{u}_{2}=R^{\theta-\pi / 2}\left(\mathbf{e}_{1}\right) | | |
| johnston-linear-matrix-algebra_FO4889 | 475 | 1.000 | \operatorname{det}\left(A^{T}\right)=0 | | |
| johnston-linear-matrix-algebra_FO4890 | 475 | 1.000 | E_{1} E_{2} \cdots E_{k} | | |
| johnston-linear-matrix-algebra_FO4891 | 475 | 1.000 | \operatorname{det}(A)=\operatorname{det}\left(E_{1} E_{2} \cdots E_{k}\right)= | | |
| johnston-linear-matrix-algebra_FO4892 | 475 | 1.000 | \operatorname{det}\left(E_{1}\right) \cdots \operatorname{det}\left(E_{k}\right) | | |
| johnston-linear-matrix-algebra_FO4893 | 475 | 1.000 | A^{T}=E_{k}^{T} \cdots E_{2}^{T} E_{1}^{T} | | |
| johnston-linear-matrix-algebra_FO4894 | 475 | 1.000 | \operatorname{det}\left(A^{T}\right)=\operatorname{det}\left(E_{k}^{T} \cdots E_{2}^{T} E_{1}^{T}\right)= | | |
| johnston-linear-matrix-algebra_FO4895 | 475 | 1.000 | \operatorname{det}\left(E_{k}^{T}\right) \cdots \operatorname{det}\left(E_{2}^{T}\right) \operatorname{det}\left(E_{1}^{T}\right) | | |
| johnston-linear-matrix-algebra_FO4896 | 475 | 1.000 | \operatorname{det}(E)=\operatorname{det}\left(E^{T}\right) | | |
| johnston-linear-matrix-algebra_FO4897 | 475 | 1.000 | E^{T}=E | | |
| johnston-linear-matrix-algebra_FO4898 | 475 | 1.000 | \operatorname{det}\left(E^{T}\right)=\operatorname{det}(E) | | |
| johnston-linear-matrix-algebra_FO4899 | 475 | 1.000 | \operatorname{det}(C) | | |
| johnston-linear-matrix-algebra_FO4900 | 476 | 1.000 | \widetilde{B} | | |
| johnston-linear-matrix-algebra_FO4901 | 476 | 1.000 | \operatorname{det}\left(I_{m}+A B\right)=\operatorname{det}\left(I_{n}+B A\right) | | |
| johnston-linear-matrix-algebra_FO4902 | 476 | 1.000 | \operatorname{det}\left(I_{n}+\mathbf{v w}^{T}\right)=1+\mathbf{w}^{T} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4903 | 477 | 1.000 | \mathbf{v}=(5,-4) | | |
| johnston-linear-matrix-algebra_FO4904 | 477 | 1.000 | \mathbf{v}=(3,2,0) | | |
| johnston-linear-matrix-algebra_FO4905 | 477 | 1.000 | \mathbf{v}=(-1,1,0,1) | | |
| johnston-linear-matrix-algebra_FO4906 | 477 | 1.000 | (A-0 I) \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4907 | 477 | 1.000 | v_{1}=-2 v_{2} | | |
| johnston-linear-matrix-algebra_FO4908 | 477 | 1.000 | \mathbf{v}=v_{2}(-2,1) | | |
| johnston-linear-matrix-algebra_FO4909 | 477 | 1.000 | \{(-2,1)\} | | |
| johnston-linear-matrix-algebra_FO4910 | 477 | 1.000 | \mathbf{v}=v_{2}(-1,1) | | |
| johnston-linear-matrix-algebra_FO4911 | 477 | 1.000 | \{(-1,1)\} | | |
| johnston-linear-matrix-algebra_FO4912 | 477 | 1.000 | \{(1,0)\} | | |
| johnston-linear-matrix-algebra_FO4913 | 477 | 1.000 | \left\{\mathbf{e}_{1}\right\},\left\{\mathbf{e}_{2}\right\} | | |
| johnston-linear-matrix-algebra_FO4914 | 477 | 1.000 | \{(1,0,0,0)\},\{(1,-5,0,0)\} | | |
| johnston-linear-matrix-algebra_FO4915 | 477 | 0.998 | \{(1,-1,-2,0)\} | | |
| johnston-linear-matrix-algebra_FO4916 | 477 | 1.000 | \{(1,1,-1,-1)\},\{(2,-1,2,-1)\} | | |
| johnston-linear-matrix-algebra_FO4917 | 477 | 1.000 | \{(1,1,0,-1),(0,1,-2,0)\} | | |
| johnston-linear-matrix-algebra_FO4918 | 477 | 1.000 | \{(0,0,1,0,-1),(1,0,0,2,0)\} | | |
| johnston-linear-matrix-algebra_FO4919 | 477 | 0.998 | \{(1,-1,-1,0,1),(2,-2,0,1,0)\} | | |
| johnston-linear-matrix-algebra_FO4920 | 477 | 0.999 | \{(0,2,2,-1,0)\} | | |
| johnston-linear-matrix-algebra_FO4921 | 477 | 1.000 | \operatorname{det}(A-\lambda I)=(-1)^{n} \operatorname{det}(\lambda I-A) | | |
| johnston-linear-matrix-algebra_FO4922 | 477 | 1.000 | \lambda=\frac{1}{2}(1+k) \pm \frac{1}{2} \sqrt{(1+k)^{2}-4 k-4}= | | |
| johnston-linear-matrix-algebra_FO4923 | 477 | 1.000 | \frac{1}{2}(1+k) \pm \frac{1}{2} \sqrt{k^{2}-2 k-3} | | |
| johnston-linear-matrix-algebra_FO4924 | 477 | 1.000 | k^{2}-2 k-3>0 | | |
| johnston-linear-matrix-algebra_FO4925 | 477 | 1.000 | k^{2}-2 k-3 | | |
| johnston-linear-matrix-algebra_FO4926 | 477 | 1.000 | (k-3)(k+1) | | |
| johnston-linear-matrix-algebra_FO4927 | 477 | 1.000 | k>3 | | |
| johnston-linear-matrix-algebra_FO4928 | 477 | 1.000 | k<-1 | | |
| johnston-linear-matrix-algebra_FO4929 | 477 | 1.000 | k=3 | | |
| johnston-linear-matrix-algebra_FO4930 | 477 | 1.000 | k=-1 | | |
| johnston-linear-matrix-algebra_FO4931 | 477 | 1.000 | -1<k<3 | | |
| johnston-linear-matrix-algebra_FO4932 | 477 | 1.000 | p_{A}(\lambda)=\lambda^{2}-\operatorname{tr}(A) \lambda+\operatorname{det}(A) | | |
| johnston-linear-matrix-algebra_FO4933 | 478 | 1.000 | \lambda= \pm 1 | | |
| johnston-linear-matrix-algebra_FO4934 | 478 | 1.000 | \lambda= \pm i | | |
| johnston-linear-matrix-algebra_FO4935 | 478 | 1.000 | A^{4} \mathbf{v}=\lambda^{4} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4936 | 478 | 0.999 | A^{4} \mathbf{v}=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4937 | 478 | 0.999 | \lambda^{4} \mathbf{v}=\mathbf{v} | | |
| johnston-linear-matrix-algebra_FO4938 | 478 | 0.999 | \lambda^{4}=1 | | |
| johnston-linear-matrix-algebra_FO4939 | 478 | 1.000 | \operatorname{det}\left(\left[R^{\theta}\right]-\lambda I\right)=0 | | |
| johnston-linear-matrix-algebra_FO4940 | 478 | 1.000 | \lambda=\cos (\theta) \pm \sqrt{\cos ^{2}(\theta)-1} | | |
| johnston-linear-matrix-algebra_FO4941 | 478 | 1.000 | \lambda=\cos (\theta) \pm | | |
| johnston-linear-matrix-algebra_FO4942 | 478 | 0.978 | \sqrt{\cos ^{2}(\theta)-1} | | |
| johnston-linear-matrix-algebra_FO4943 | 478 | 0.999 | \cos ^{2}(\theta)-1 \geq 0 | | |
| johnston-linear-matrix-algebra_FO4944 | 478 | 0.999 | \cos (\theta)= \pm 1 | | |
| johnston-linear-matrix-algebra_FO4945 | 478 | 1.000 | \theta=k \pi | | |
| johnston-linear-matrix-algebra_FO4946 | 478 | 1.000 | \theta=2 k \pi | | |
| johnston-linear-matrix-algebra_FO4947 | 478 | 0.999 | \theta=(2 k+1) \pi | | |
| johnston-linear-matrix-algebra_FO4948 | 478 | 1.000 | \theta \neq k \pi | | |
| johnston-linear-matrix-algebra_FO4949 | 478 | 1.000 | \mathbf{v} \cdot \mathbf{w}=\mathbf{v}^{*} \mathbf{w} | | |
| johnston-linear-matrix-algebra_FO4950 | 478 | 1.000 | \mathbf{x}^{*} A \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4951 | 478 | 1.000 | \left(A^{*} \mathbf{x}\right) \cdot \mathbf{y}=\left(A^{*} \mathbf{x}\right)^{*} \mathbf{y}=\mathbf{x}^{*} A \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4952 | 478 | 1.000 | \mathbf{x} \cdot(A \mathbf{y})=\left(A^{*} \mathbf{x}\right) \cdot \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4953 | 478 | 1.000 | \mathbf{x} \cdot(A \mathbf{y})=\mathbf{x}^{*} A \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4954 | 478 | 1.000 | (B \mathbf{x}) \cdot \mathbf{y}=\mathbf{x}^{*} B^{*} \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4955 | 478 | 1.000 | \mathbf{x}^{*} A \mathbf{y}= | | |
| johnston-linear-matrix-algebra_FO4956 | 478 | 0.999 | \mathbf{x}^{*} B^{*} \mathbf{y} | | |
| johnston-linear-matrix-algebra_FO4957 | 478 | 0.999 | \mathbf{x} \in \mathbb{C}^{m} | | |
| johnston-linear-matrix-algebra_FO4958 | 478 | 0.999 | \mathbf{y} \in \mathbb{C}^{n} | | |
| johnston-linear-matrix-algebra_FO4959 | 478 | 1.000 | \mathbf{x}^{*} A \mathbf{y}=a_{i, j} | | |
| johnston-linear-matrix-algebra_FO4960 | 478 | 1.000 | \mathbf{x}^{*} B^{*} \mathbf{y}=b_{j, i} | | |
| johnston-linear-matrix-algebra_FO4961 | 478 | 1.000 | a_{i, j}=\overline{b_{j, i}} | | |
| johnston-linear-matrix-algebra_FO4962 | 478 | 0.975 | A \mathbf{v}=0 \mathbf{v}=\mathbf{0} | | |
| johnston-linear-matrix-algebra_FO4963 | 478 | 1.000 | \left(A^{*}\right)^{*}=A | | |
| johnston-linear-matrix-algebra_FO4964 | 478 | 1.000 | \overline{\bar{\lambda}}=\lambda | | |
| johnston-linear-matrix-algebra_FO4965 | 478 | 1.000 | \mathbf{w} \cdot(A \mathbf{v}) | | |
| johnston-linear-matrix-algebra_FO4966 | 478 | 1.000 | \mathbf{w} \cdot \mathbf{v}=0 | | |
| johnston-linear-matrix-algebra_FO4967 | 478 | 1.000 | \lambda_{2}, \lambda_{3}, \ldots, \lambda_{n} | | |
| johnston-linear-matrix-algebra_FO4968 | 478 | 1.000 | \overline{\lambda_{j}} | | |
| johnston-linear-matrix-algebra_FO4969 | 478 | 1.000 | \lambda_{1} \neq | | |
| johnston-linear-matrix-algebra_FO4970 | 478 | 1.000 | \lambda_{1}=\lambda_{j} | | |
| johnston-linear-matrix-algebra_FO4971 | 478 | 1.000 | B^{*} | | |
| johnston-linear-matrix-algebra_FO4972 | 478 | 1.000 | \overline{\lambda_{2}}, \overline{\lambda_{3}}, \ldots, \overline{\lambda_{n}} | | |
| johnston-linear-matrix-algebra_FO4973 | 478 | 1.000 | \lambda_{1}=1 | | |
| johnston-linear-matrix-algebra_FO4974 | 478 | 1.000 | \lambda_{2}=2 | | |
| johnston-linear-matrix-algebra_FO4975 | 478 | 1.000 | \mathbf{v}_{1}=(1,0) | | |
| johnston-linear-matrix-algebra_FO4976 | 478 | 1.000 | v_{2}=(1,1) / \sqrt{2} | | |
| johnston-linear-matrix-algebra_FO4977 | 478 | 1.000 | \mathbf{v}_{2}=(1,2) | | |
| johnston-linear-matrix-algebra_FO4978 | 479 | 1.000 | A \mathbf{1}=\mathbf{1} | | |
| johnston-linear-matrix-algebra_FO4979 | 479 | 1.000 | A \mathbf{1} | | |
| johnston-linear-matrix-algebra_FO4980 | 479 | 0.973 | \left|v_{j}\right| \leq 1 | | |
| johnston-linear-matrix-algebra_FO4981 | 479 | 1.000 | \lambda=i | | |
| johnston-linear-matrix-algebra_FO4982 | 480 | 1.000 | 2+i, 2+i, 2-i, 2-i | | |
| johnston-linear-matrix-algebra_FO4983 | 480 | 0.999 | \left[\begin{array}{ll}\sin (2) / 2 & \sin (2) / 2 \\ \sin (2) / 2 & \sin (2) / 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO4984 | 480 | 1.000 | P=D=O | | |
| johnston-linear-matrix-algebra_FO4985 | 480 | 1.000 | A=P D P^{-1}=P(\lambda I) P^{-1}=\lambda P P^{-1}=\lambda I | | |
| johnston-linear-matrix-algebra_FO4986 | 480 | 1.000 | L_{0}=2 | | |
| johnston-linear-matrix-algebra_FO4987 | 480 | 0.999 | 2 \phi-1=\sqrt{5} | | |
| johnston-linear-matrix-algebra_FO4988 | 480 | 1.000 | \operatorname{rank}(A)=\operatorname{rank}\left(P D P^{-1}\right)=\operatorname{rank}\left(D P^{-1}\right)= | | |
| johnston-linear-matrix-algebra_FO4989 | 480 | 0.996 | \operatorname{rank}(D) | | |
| johnston-linear-matrix-algebra_FO4990 | 480 | 1.000 | B=P D P^{-1} | | |
| johnston-linear-matrix-algebra_FO4991 | 481 | 0.913 | (\{(1,0)\} | | |
| johnston-linear-matrix-algebra_FO4992 | 481 | 1.000 | P \widetilde{D} P^{-1} | | |
| johnston-linear-matrix-algebra_FO4993 | 481 | 1.000 | k^{n} k | | |
| johnston-linear-matrix-algebra_FO4994 | 481 | 1.000 | P= | | |
| johnston-linear-matrix-algebra_FO4995 | 481 | 1.000 | \left[\mathbf{v}_{1}\left|\mathbf{v}_{2}\right| P_{2}\right] | | |
| johnston-linear-matrix-algebra_FO4996 | 481 | 1.000 | \left\{c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}, d_{1} \mathbf{v}_{1}+d_{2} \mathbf{v}_{2}\right\} | | |
| johnston-linear-matrix-algebra_FO4997 | 481 | 1.000 | A=Q D Q^{-1} | | |
| johnston-linear-matrix-algebra_FO4998 | 481 | 1.000 | \widetilde{D} | | |
| johnston-linear-matrix-algebra_FO4999 | 481 | 1.000 | Q \widetilde{D} Q^{-1} | | |
| johnston-linear-matrix-algebra_FO5000 | 481 | 1.000 | c_{1}, c_{2}, d_{1}, d_{2} | | |
| johnston-linear-matrix-algebra_FO5001 | 481 | 1.000 | A^{2} \mathbf{v}=\lambda A \mathbf{v}=\lambda^{2} \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO5002 | 481 | 1.000 | \lambda^{0}=1 | | |
| johnston-linear-matrix-algebra_FO5003 | 481 | 1.000 | p(x)=c_{k} x^{k}+\cdots+c_{1} x+c_{0} | | |
| johnston-linear-matrix-algebra_FO5004 | 481 | 1.000 | x^{r} x^{s}=x^{r+s} | | |
| johnston-linear-matrix-algebra_FO5005 | 481 | 1.000 | \left(x^{r}\right)^{s}=x^{r s} | | |
| johnston-linear-matrix-algebra_FO5006 | 481 | 1.000 | \operatorname{det}\left(e^{A+B}\right)=e^{\operatorname{tr}(A+B)} | | |
| johnston-linear-matrix-algebra_FO5007 | 481 | 1.000 | -a_{0}-a_{1} \lambda_{j}-a_{2} \lambda_{j}^{2}- | | |
| johnston-linear-matrix-algebra_FO5008 | 481 | 1.000 | \cdots-a_{n-1} \lambda_{j}^{n-1} | | |
| johnston-linear-matrix-algebra_FO5009 | 481 | 1.000 | \lambda_{j}^{n} | | |
| johnston-linear-matrix-algebra_FO5010 | 482 | 1.000 | C \mathbf{v}=\lambda \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO5011 | 482 | 1.000 | \operatorname{tr}(B)= | | |
| johnston-linear-matrix-algebra_FO5012 | 482 | 1.000 | (\lambda-2)(\lambda-3) | | |
| johnston-linear-matrix-algebra_FO5013 | 482 | 1.000 | (\lambda-1)(\lambda-3)^{2} | | |
| johnston-linear-matrix-algebra_FO5014 | 482 | 1.000 | A^{2} \mathbf{v}=3 A \mathbf{v}=9 \mathbf{v} | | |
| johnston-linear-matrix-algebra_FO5015 | 482 | 1.000 | p_{A}(\lambda)=p_{B}(\lambda)=\lambda^{4} | | |
| johnston-linear-matrix-algebra_FO5016 | 482 | 0.728 | \operatorname{span}\{(1,0,0,0),(0,1,0,0)\} | | |
| johnston-linear-matrix-algebra_FO5017 | 482 | 1.000 | v_{2}=\lambda v_{1} | | |
| johnston-linear-matrix-algebra_FO5018 | 482 | 1.000 | v_{3}=\lambda v_{2} | | |
| johnston-linear-matrix-algebra_FO5019 | 482 | 1.000 | v_{n}=\lambda v_{n-1} | | |
| johnston-linear-matrix-algebra_FO5020 | 482 | 1.000 | v_{j}=\lambda^{j-1} v_{1} | | |
| johnston-linear-matrix-algebra_FO5021 | 482 | 1.000 | 1 \leq j \leq n-1 | | |
| johnston-linear-matrix-algebra_FO5022 | 482 | 1.000 | v_{n}=\lambda^{n-1} v_{1} | | |
| johnston-linear-matrix-algebra_FO5023 | 482 | 1.000 | v_{1}\left(1, \lambda, \lambda^{2}, \ldots, \lambda^{n-1}\right) | | |
| johnston-linear-matrix-algebra_FO5024 | 482 | 1.000 | P \in \mathcal{M}_{4} | | |
| johnston-linear-matrix-algebra_FO5025 | 482 | 1.000 | a+d=0 | | |
| johnston-linear-matrix-algebra_FO5026 | 482 | 1.000 | b(0+0)=1 | | |
| johnston-linear-matrix-algebra_FO5027 | 483 | 0.979 | \operatorname{det}(A)=-3, \operatorname{det}\left(A_{1}\right)=-3, \operatorname{det}\left(A_{2}\right)=-3 | | |
| johnston-linear-matrix-algebra_FO5028 | 483 | 0.971 | \operatorname{det}(A)=4, \quad \operatorname{det}\left(A_{1}\right)=2, \quad \operatorname{det}\left(A_{2}\right)=8 | | |
| johnston-linear-matrix-algebra_FO5029 | 483 | 1.000 | \operatorname{det}\left(A_{3}\right)=6,(x, y, z)=(1 / 2,2,3 / 2) | | |
| johnston-linear-matrix-algebra_FO5030 | 483 | 0.998 | \operatorname{det}(A)=-2, \operatorname{det}\left(A_{1}\right)=1, \operatorname{det}\left(A_{2}\right)=-2 | | |
| johnston-linear-matrix-algebra_FO5031 | 483 | 1.000 | \operatorname{det}\left(A_{3}\right)=-1,(x, y, z)=(-1 / 2,1,1 / 2) | | |
| johnston-linear-matrix-algebra_FO5032 | 483 | 0.999 | \operatorname{rank}(\operatorname{cof}(A))=n | | |
| johnston-linear-matrix-algebra_FO5033 | 483 | 0.741 | \boldsymbol{l \circ \boldsymbol { l } = \boldsymbol { l } \text { . }} | | |
| johnston-linear-matrix-algebra_FO5034 | 483 | 1.000 | n=5 | | |
| johnston-linear-matrix-algebra_FO5035 | 483 | 1.000 | 5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120 | | |
| johnston-linear-matrix-algebra_FO5036 | 483 | 1.000 | \operatorname{cof}\left(A^{2}\right)=\operatorname{cof}(A A)=\operatorname{cof}(A) \operatorname{cof}(A)=(\operatorname{cof}(A))^{2} | | |
| johnston-linear-matrix-algebra_FO5037 | 483 | 1.000 | \operatorname{cof}\left(A^{-1}\right)= | | |
| johnston-linear-matrix-algebra_FO5038 | 483 | 1.000 | A^{T} / \operatorname{det}(A)=(\operatorname{cof}(A))^{-1} | | |
| johnston-linear-matrix-algebra_FO5039 | 483 | 1.000 | \operatorname{cof}(A)= | | |
| johnston-linear-matrix-algebra_FO5040 | 483 | 1.000 | \operatorname{det}(A)\left(A^{T}\right)^{-1} | | |
| johnston-linear-matrix-algebra_FO5041 | 483 | 1.000 | \tau=(21345 \cdots n) | | |
| johnston-linear-matrix-algebra_FO5042 | 483 | 1.000 | \operatorname{sgn}(\tau)=-1 | | |
| johnston-linear-matrix-algebra_FO5043 | 483 | 0.993 | \operatorname{rank}(A) \leq n<m | | |
| johnston-linear-matrix-algebra_FO5044 | 483 | 0.993 | \operatorname{rank}(A B) \leq n<m | | |
| johnston-linear-matrix-algebra_FO5045 | 483 | 1.000 | \operatorname{det}(A B)=0 | | |
| johnston-linear-matrix-algebra_FO5046 | 483 | 0.999 | \operatorname{per}(A)=2, \operatorname{per}(B)=10 | | |
| johnston-linear-matrix-algebra_FO5047 | 483 | 0.999 | \operatorname{per}(A B)=48 | | |
| johnston-linear-matrix-algebra_FO5048 | 484 | 0.881 | 1 \quad | | |
| johnston-linear-matrix-algebra_FO5049 | 484 | 0.881 | \quad \lambda \approx 5.37, \mathbf{v} \approx(0.42,0.91) | | |
| johnston-linear-matrix-algebra_FO5050 | 484 | 1.000 | \lambda \approx-3.33, \mathbf{v} \approx(0.73,-0.42,-0.55) | | |
| johnston-linear-matrix-algebra_FO5051 | 484 | 1.000 | \lambda \approx 7.92, \mathbf{v} \approx(0.73,0.29,0.57,0.24) | | |
| johnston-linear-matrix-algebra_FO5052 | 484 | 1.000 | 2 \quad | | |
| johnston-linear-matrix-algebra_FO5053 | 484 | 1.000 | \quad \lambda_{1} \approx 5.37, \mathbf{v}_{1} \approx(0.42,0.91) | | |
| johnston-linear-matrix-algebra_FO5054 | 484 | 1.000 | \lambda_{2} \approx-0.37, \mathbf{v}_{2} \approx(0.82,-0.57) | | |
| johnston-linear-matrix-algebra_FO5055 | 484 | 1.000 | \lambda_{1} \approx 16.12, \mathbf{v}_{1} \approx(0.23,0.53,0.82) | | |
| johnston-linear-matrix-algebra_FO5056 | 484 | 1.000 | \lambda_{2} \approx-1.12, \mathbf{v}_{2} \approx(0.79,0.09,-0.61) | | |
| johnston-linear-matrix-algebra_FO5057 | 484 | 1.000 | \lambda_{3} \approx 0.00, \mathbf{v}_{2} \approx(0.41,-0.82,0.41) | | |
| johnston-linear-matrix-algebra_FO5058 | 484 | 0.520 | 3 \quad | | |
| johnston-linear-matrix-algebra_FO5059 | 484 | 0.520 | \lambda \approx 11.58 | | |
| johnston-linear-matrix-algebra_FO5060 | 484 | 1.000 | \mathbf{v} \approx(0.15,0.25,0.01,0.54,-0.27,0.46,0.58) | | |
| johnston-linear-matrix-algebra_FO5061 | 484 | 1.000 | A \mathbf{v}_{k} \approx \lambda_{1} \mathbf{v}_{k} | | |
| johnston-linear-matrix-algebra_FO5062 | 484 | 1.000 | \left\|A \mathbf{v}_{k}\right\| \approx\left\|\lambda_{1} \mathbf{v}_{k}\right\|=\lambda_{1} | | |
| johnston-linear-matrix-algebra_FO5063 | 484 | 0.994 | k-2 \cdot 4^{2}- | | |
| johnston-linear-matrix-algebra_FO5064 | 484 | 1.000 | k=4: A^{4}=\left[\begin{array}{ll}8 & 8 \\ 8 & 8\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5065 | 484 | 0.993 | k=4: A^{4}=\left[\begin{array}{lll}5 & 4 & 1 \\ 4 & 6 & 3 \\ 1 & 3 & 2\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5066 | 484 | 0.445 | \lambda \approx 1.62, \mathbf{v} \approx(0.53,0.85) | | |
| johnston-linear-matrix-algebra_FO5067 | 484 | 1.000 | \lambda \approx 2.00 i, \mathbf{v} \approx(0.71,0.71) | | |
| johnston-linear-matrix-algebra_FO5068 | 484 | 1.000 | \lambda \approx 1.80, \mathbf{v} \approx(0.59,0.74,0.33) | | |
| johnston-linear-matrix-algebra_FO5069 | 484 | 1.000 | \left[R^{\pi / 2}\right] | | |
| johnston-linear-matrix-algebra_FO5070 | 484 | 1.000 | Q=I, r=1 | | |
| johnston-linear-matrix-algebra_FO5071 | 484 | 1.000 | 1 \pm i=\sqrt{2} e^{ \pm i \pi / 4} | | |
| johnston-linear-matrix-algebra_FO5072 | 484 | 0.999 | (1,-1 \pm i) | | |
| johnston-linear-matrix-algebra_FO5073 | 484 | 1.000 | r=\sqrt{2}, \theta=\pi / 4 | | |
| johnston-linear-matrix-algebra_FO5074 | 484 | 0.922 | B=\operatorname{diag}\left(1,\left[R^{\pi / 2}\right]\right), Q=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & -1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5075 | 484 | 1.000 | B=\operatorname{diag}\left(1,2, \sqrt{2}\left[R^{\pi / 4}\right]\right) | | |
| johnston-linear-matrix-algebra_FO5076 | 484 | 0.998 | B=\operatorname{diag}\left(2 \sqrt{2}\left[R^{\pi / 4}\right], 2 \sqrt{2}\left[R^{\pi / 4}\right], 4\right) | | |
| johnston-linear-matrix-algebra_FO5077 | 485 | 0.391 | 2^{k}(\cos (k \pi / 6) \quad-\sin (k \pi / 6)) | | |
| johnston-linear-matrix-algebra_FO5078 | 485 | 1.000 | 2^{k} \sin (k \pi / 6) | | |
| johnston-linear-matrix-algebra_FO5079 | 485 | 1.000 | -2^{k+1} \sin (k \pi / 6) | | |
| johnston-linear-matrix-algebra_FO5080 | 485 | 1.000 | 2^{k}(\cos (k \pi / 6)+\sin (k \pi / 6)) | | |
| johnston-linear-matrix-algebra_FO5081 | 485 | 1.000 | \lambda^{2}-\lambda-6 | | |
| johnston-linear-matrix-algebra_FO5082 | 485 | 0.998 | \lambda_{0}=3 | | |
| johnston-linear-matrix-algebra_FO5083 | 485 | 0.998 | \lambda_{1}=-2 | | |
| johnston-linear-matrix-algebra_FO5084 | 485 | 1.000 | 3 \pm \sqrt{5} | | |
| johnston-linear-matrix-algebra_FO5085 | 485 | 0.999 | 1 \pm \sqrt{3} | | |
| johnston-linear-matrix-algebra_FO5086 | 485 | 1.000 | \left[\begin{array}{ll}0 & 1 \\ 6 & 1\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5087 | 485 | 1.000 | \left[\begin{array}{cc}0 & 1 \\ -4 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5088 | 485 | 1.000 | \left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & -11 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5089 | 485 | 1.000 | \left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ 8 & -12 & 6\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5090 | 485 | 1.000 | \left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & -4 & 5\end{array}\right] | | |
| johnston-linear-matrix-algebra_FO5091 | 485 | 1.000 | \lambda^{2} z-\lambda-6 | | |
| johnston-linear-matrix-algebra_FO5092 | 485 | 1.000 | c_{0} | | |
| johnston-linear-matrix-algebra_FO5093 | 485 | 1.000 | x_{n}=c_{0} 3^{n}+c_{1}(-2)^{n} | | |
| johnston-linear-matrix-algebra_FO5094 | 485 | 1.000 | x_{0}=4 | | |
| johnston-linear-matrix-algebra_FO5095 | 485 | 1.000 | x_{1}=-3 | | |
| johnston-linear-matrix-algebra_FO5096 | 485 | 1.000 | x_{n}=3^{n}+ | | |
| johnston-linear-matrix-algebra_FO5097 | 485 | 1.000 | 3(-2)^{n} | | |
| johnston-linear-matrix-algebra_FO5098 | 485 | 1.000 | x_{n}=i^{n}+(-i)^{n} | | |
| johnston-linear-matrix-algebra_FO5099 | 485 | 1.000 | x_{n}= | | |
| johnston-linear-matrix-algebra_FO5100 | 485 | 1.000 | 2 \cos (\pi n / 2) | | |
| johnston-linear-matrix-algebra_FO5101 | 485 | 1.000 | x_{n}=3^{n}-1 | | |
| johnston-linear-matrix-algebra_FO5102 | 485 | 1.000 | x_{n}=(2 n-3)+2^{n} | | |
| johnston-linear-matrix-algebra_FO5103 | 485 | 1.000 | \lambda^{2}+1 | | |
| johnston-linear-matrix-algebra_FO5104 | 485 | 1.000 | \pm i | | |
| johnston-linear-matrix-algebra_FO5105 | 485 | 1.000 | n^{2}-n^{3} | | |
| johnston-linear-matrix-algebra_FO5106 | 485 | 1.000 | 4-1=3 | | |
| johnston-linear-matrix-algebra_FO5107 | 485 | 1.000 | x_{n}=4 x_{n-1}-6 x_{n-2}+ | | |
| johnston-linear-matrix-algebra_FO5108 | 485 | 1.000 | 4 x_{n-3}-x_{n-4} | | |
| johnston-linear-matrix-algebra_FO5109 | 485 | 1.000 | (\lambda-1)^{4} | | |
| johnston-linear-matrix-algebra_FO5110 | 485 | 0.998 | p(\lambda)=(\lambda-2)(\lambda-3)=\lambda^{2}-5 \lambda+6 | | |
| johnston-linear-matrix-algebra_FO5111 | 485 | 1.000 | x_{n}=9 x_{n-1}-26 x_{n-2}+24 x_{n-3} | | |
| johnston-linear-matrix-algebra_FO5112 | 485 | 1.000 | x_{n}=9 x_{n-1}-30 x_{n-2}+44 x_{n-3}-24 x_{n-4} | | |
| johnston-linear-matrix-algebra_FO5113 | 485 | 1.000 | x_{n}=3 x_{n-1}-3 x_{n-2}+x_{n-3} | | |
| johnston-linear-matrix-algebra_FO5114 | 485 | 1.000 | x_{n}=8 x_{n-1}-26 x_{n-2}+48 x_{n-3}-45 x_{n-4} | | |
| johnston-linear-matrix-algebra_FO5115 | 485 | 1.000 | \pm i= | | |
| johnston-linear-matrix-algebra_FO5116 | 485 | 1.000 | e^{ \pm i \pi / 2} | | |
| johnston-linear-matrix-algebra_FO5117 | 485 | 1.000 | ( \pm i)^{n}=\left(e^{ \pm i \pi / 2}\right)^{n}= | | |
| johnston-linear-matrix-algebra_FO5118 | 485 | 1.000 | e^{ \pm i n \pi / 2} | | |
| johnston-linear-matrix-algebra_FO5119 | 485 | 1.000 | x_{n}=1+2^{n+1} \cos (n \pi / 2) | | |
| johnston-linear-matrix-algebra_FO5120 | 485 | 1.000 | x_{0}=x_{1}=0 | | |
| johnston-linear-matrix-algebra_FO5121 | 485 | 1.000 | x_{n-1}=x_{n-2} | | |
| johnston-linear-matrix-algebra_FO5122 | 485 | 1.000 | x_{n+1}=x_{n}-4 x_{n-1}+4 x_{n-2}= | | |
| johnston-linear-matrix-algebra_FO5123 | 485 | 1.000 | x_{n}-4 x_{n-1}+4 x_{n-1}=x_{n} | |