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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|0KCQ9Y3QKA]] $B = (b_1, \\dots, b_n)$ and $C = (c_1, \\dots, c_m)$.\nLet $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]].\nThe *matrix of $T$* in the bases $B$ and $C$ is the $m \\times n$ matrix $[T]_{C,B} = (a_{ij})$ defined by\n$$T(b_j) = \\sum_{i=1}^{m} a_{ij} c_i \\qquad (1 \\le j \\le n).$$\nEquivalently, column $j$ of $[T]_{C,B}$ is the [[coordinate vector|04CYQHN6MC]] $[T(b_j)]_C$.\n\n## Remarks\n\nFor each $v \\in V$, $[T(v)]_C = [T]_{C,B} [v]_B$.\nThe map $T \\mapsto [T]_{C,B}$ is a linear isomorphism $\\Hom_F(V, W) \\to M_{m \\times n}(F)$.\nFor the rotation of $\\R^2$ by an angle $\\theta$, in the standard basis, the matrix is $\\begin{pmatrix} \\cos\\theta \u0026 -\\sin\\theta \\\\ \\sin\\theta \u0026 \\cos\\theta \\end{pmatrix}$.\nMathlib writes this map as `LinearMap.toMatrix`.\n\n## Explanations\n\n### L1 human\n\nTo write a linear map as a matrix, apply the map to each basis vector.\nWrite the coordinates of each image as a column.\nThese columns, side by side, are the matrix.\n",
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