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  "body": "\n## Definition\n\nLet $K$ be a field. Let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space $V$, and let $B$ be an affine space over a $K$-vector space $W$.\nA map $f : A \\to B$ is an affine map if there is a linear map $L : V \\to W$ such that\n$$f(p + v) = f(p) + L(v) \\quad \\text{for each point } p \\text{ of } A \\text{ and each vector } v \\text{ of } V.$$\nThe map $L$ is unique. It is the linear part of $f$, and it satisfies $L(q - p) = f(q) - f(p)$ for all points $p$ and $q$ of $A$.\n\n## Remarks\n\nThe affine maps from $\\mathbb{R}^n$ to $\\mathbb{R}^m$ are the maps $x \\mapsto M x + b$, with $M$ a real $m \\times n$ matrix and $b$ a vector of $\\mathbb{R}^m$.\nAn affine map keeps [[affine combinations|0KJH6SRFZK]]. In particular it sends [[midpoints|0S706JBZBG]] to midpoints.\nAn affine transformation is a bijective affine map from $A$ to itself. The affine transformations of $A$ form a group under composition.\nMathlib's `AffineMap` is this notion, with the linear part as a field of the structure.\n\n## Explanations\n\n### L1 human\n\nAn affine map is a linear map that may also shift the origin. In the plane, examples are rotations, stretches, shears and translations, and the compositions of these. A grid of squares goes to a grid of parallelograms.\n",
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