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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ be an $m \\times n$ matrix over $F$.\nThe *linear map of multiplication by $A$* is\n$$L_A \\colon F^n \\to F^m, \\qquad L_A(x) = Ax,$$\nwhere $Ax$ is the [[matrix–vector product|0PDVQEP0ZC]].\n\n## Remarks\n\nThe map $L_A$ is linear, because $A(x + y) = Ax + Ay$ and $A(cx) = c\\, Ax$.\nIts matrix in the standard bases is $A$, because $L_A(e_j)$ is column $j$ of $A$.\nEach linear map $F^n \\to F^m$ equals $L_A$ for exactly one matrix $A$.\nThe kernel of $L_A$ is the [[null space|0PHDYNR24D]] of $A$, and its image is the [[column space|03AJKXH1AV]] of $A$.\n\n## Explanations\n\n### L1 human\n\nMultiplying vectors by a fixed matrix is a linear map.\nThe matrix $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 3 \\end{pmatrix}$ stretches the $x$-direction by $2$ and the $y$-direction by $3$.\nEvery linear map between coordinate spaces is of this kind.\n",
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