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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $P$ be an [[affine space|09XNAPTJQ7]] over $V$.\nThe *affine group* $\\operatorname{Aff}(P)$ is the set of all bijective [[affine maps|0JF0TVYGWT]] $P \\to P$, with composition as the group operation.\nFor $P = V$ it is written $\\operatorname{Aff}(V)$, and for $V = F^n$ it is written $\\operatorname{AGL}(n, F)$.\n\n## Remarks\n\nExample: on $\\R^2$, the map $f(x) = Ax + b$ with $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ and $b = (1, 0)$ is in $\\operatorname{AGL}(2, \\R)$, since $\\det A = 1 \\ne 0$; its inverse is $x \\mapsto A^{-1}(x - b)$.\nNon-example: $f(x) = Ax + b$ with $A = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$ is affine but not bijective, so it is not in the affine group.\nEach element of $\\operatorname{AGL}(n, F)$ has the form $x \\mapsto Ax + b$ with $A \\in \\operatorname{GL}(n, F)$ and $b \\in F^n$, in a unique way.\nThe map to the linear part $A$ is a surjective homomorphism onto the [[general linear group|0FCHE902D7]], whose kernel is the group of [[translations|087VJM5Q77]], isomorphic to $F^n$.\nHence $\\operatorname{AGL}(n, F) \\cong F^n \\rtimes \\operatorname{GL}(n, F)$, a [[semidirect product|0H3WZXCZNF]].\nIt is isomorphic to the group of $(n+1) \\times (n+1)$ matrices $\\begin{pmatrix} A \u0026 b \\\\ 0 \u0026 1 \\end{pmatrix}$ with $A \\in \\operatorname{GL}(n, F)$.\n",
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