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  "body": "\n## Definition\n\nLet $F$ be a field, and let $V$ and $W$ be vector spaces over $F$ with bases $B = (b_1, \\dots, b_n)$ and $C = (c_1, \\dots, c_m)$.\nLet $A = (a_{ij})$ be an $m \\times n$ matrix over $F$.\nThe *linear map of $A$* in the bases $B$ and $C$ is the linear map $T_A \\colon V \\to W$ with\n$$T_A(b_j) = \\sum_{i=1}^{m} a_{ij} c_i \\qquad (1 \\le j \\le n).$$\n\n## Remarks\n\nThe map $A \\mapsto T_A$ is the inverse of the map $T \\mapsto [T]_{C,B}$ of the [[matrix of a linear map|0YQSGNYE3K]].\nFor $V = F^n$ and $W = F^m$ with the standard bases, $T_A$ is the map $x \\mapsto Ax$, see [[multiplication by a matrix as a linear map|094V2VBJXJ]].\nIn Mathlib, this map is `Matrix.toLin`.\n\n## Explanations\n\n### L1 human\n\nA matrix and two bases together define a linear map.\nColumn $j$ tells us where the map sends basis vector $j$.\nThis undoes the step from a linear map to its matrix.\n",
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