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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ and $W$ be [[vector spaces|000AYPA9AJ]] over $F$.\nA map $T \\colon V \\to W$ is *linear* when, for all $u, v \\in V$ and $a \\in F$,\n$$T(u + v) = T(u) + T(v), \\qquad T(av) = a\\, T(v).$$\nA linear map is also called a *linear transformation*.\n\n## Remarks\n\nA linear map sends $0$ to $0$ and linear combinations to linear combinations: $T(\\sum a_i v_i) = \\sum a_i T(v_i)$.\nFor an $m \\times n$ matrix $A$, the map $F^n \\to F^m$, $x \\mapsto Ax$, is linear.\nRotations of $\\R^2$ about the origin are linear; translations by a nonzero vector are not linear.\nEach linear map between finite-dimensional spaces has a [[matrix|0YQSGNYE3K]] in a choice of bases.\n\n## Explanations\n\n### L1 human\n\nA linear map respects the two operations of a vector space.\nThe image of a sum is the sum of the images, and the image of a scaled vector is the scaled image.\nIn the plane, linear maps send lines through the origin to lines through the origin, or to the origin, and keep parallel lines parallel.\n",
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