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  "body": "\n## Summary\n\nThe affine group of an [[affine space|09XNAPTJQ7]] is the group of all its invertible [[affine maps|0JF0TVYGWT]].\nOn $F^n$ these are the maps $x \\mapsto Ax + b$ with $A$ an invertible matrix and $b$ a vector.\nIt is the [[semidirect product|0H3WZXCZNF]] of the [[translations|087VJM5Q77]] and the [[general linear group|0FCHE902D7]].\nIn the [[Erlangen program|9QW65PGEPW]], it is the group that defines affine geometry.\n\n## Explanations\n\n### L1 human\n\nAn affine map moves the plane so that straight lines stay straight and parallel lines stay parallel.\nIt may stretch, shear, rotate, reflect and shift, but it may not bend.\nFor example, $(x, y) \\mapsto (2x + y + 1, \\, y - 3)$ is an invertible affine map of the plane.\nThe affine group collects all such maps, with composition as the operation.\n",
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