线性变换的矩阵例题
2023-03-12 09:25:51
给定一个线性变换 $\\phi:\\mathbb{R}^3\\rightarrow\\mathbb{R}^2$，满足 $\\phi\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}=\\begin{pmatrix}1\\\\1\\end{pmatrix}$，$\\phi\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}=\\begin{pmatrix}-1\\\\2\\end{pmatrix}$，$\\phi\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}=\\begin{pmatrix}3\\\\1\\end{pmatrix}$。求 $\\phi$ 的矩阵表示。
解：设 $\\phi$ 的矩阵表示为 $A$，则有
$$
A\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}=\\begin{pmatrix}1\\\\1\\end{pmatrix},\\ A\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}=\\begin{pmatrix}-1\\\\2\\end{pmatrix},\\ A\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}=\\begin{pmatrix}3\\\\1\\end{pmatrix}
$$
即
$$
\\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\end{pmatrix}\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}=\\begin{pmatrix}1\\\\1\\end{pmatrix},\\ \\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\end{pmatrix}\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}=\\begin{pmatrix}-1\\\\2\\end{pmatrix},\\ \\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\end{pmatrix}\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}=\\begin{pmatrix}3\\\\1\\end{pmatrix}
$$
展开得到
$$
\\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\end{pmatrix}\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}+\\begin{pmatrix}0\\\\0\\end{pmatrix}=\\begin{pmatrix}1\\\\1\\end{pmatrix},\\ \\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\end{pmatrix}\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}+\\begin{pmatrix}0\\\\0\\end{pmatrix}=\\begin{pmatrix}-1\\\\2\\end{pmatrix},\\ \\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\end{pmatrix}\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}+\\begin{pmatrix}0\\\\0\\end{pmatrix}=\\begin{pmatrix}3\\\\1\\end{pmatrix}
$$
即
$$
\\begin{pmatrix}a_{11}\\\\a_{21}\\end{pmatrix}=\\begin{pmatrix}1\\\\1\\end{pmatrix},\\ \\begin{pmatrix}a_{12}\\\\a_{22}\\end{pmatrix}=\\begin{pmatrix}-1\\\\2\\end{pmatrix},\\ \\begin{pmatrix}a_{13}\\\\a_{23}\\end{pmatrix}=\\begin{pmatrix}3\\\\1\\end{pmatrix}
$$
$\\phi$ 的矩阵表示为
$$
A=\\begin{pmatrix}1&-1&3\\\\1&2&1\\end{pmatrix}
$$