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  "body": "\n## Statement\n\nLet $A$ be an $m \\times n$ real matrix. Let $T \\colon \\mathbb{R}^n \\to \\mathbb{R}^m$ be the [[map given by multiplication by a matrix|0AJ0KP1B31]], so that $T(x) = Ax$ for every $x$ in $\\mathbb{R}^n$.\n\nFor $j = 1, \\dots, n$, let $e_j$ be the vector of $\\mathbb{R}^n$ with entry $1$ in place $j$ and entry $0$ elsewhere.\n\nThen column $j$ of $A$ is the vector $T(e_j)$ for each $j$. In other words, $A$ is the matrix whose columns are $T(e_1), T(e_2), \\dots, T(e_n)$, in this order.\n\n## Remarks\n\nThe reason is that the [[matrix-vector product|0CCQV3884B]] $Ae_j$ picks out the $j$-th column of $A$.\n\nThe statement shows that a matrix transformation determines its matrix.\n",
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