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  "body": "\n## Statement\n\nLet $T \\colon \\mathbb{R}^n \\to \\mathbb{R}^m$ be a function. There is an $m \\times n$ matrix $A$ with $T(x) = Ax$ for all $x \\in \\mathbb{R}^n$ if and only if $T$ is a [[linear map|0NRYHJPQG2]]. That is, $T(cv) = cT(v)$ and $T(v+w) = T(v) + T(w)$ for all scalars $c$ and all vectors $v, w$.\n\nWhen this holds, the matrix $A$ is the one whose $j$-th column is $T(e_j)$, where $e_1, \\dots, e_n$ are the standard basis vectors of $\\mathbb{R}^n$. So $A = [\\,T(e_1)\\ \\ T(e_2)\\ \\cdots\\ T(e_n)\\,]$.\n\n## Remarks\n\nMultiplication by a matrix always respects sums and scalar multiples. The converse holds because each $x = \\sum_j x_j e_j$ is sent to $\\sum_j x_j T(e_j)$, which is the product of the matrix of these columns with $x$.\n",
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