Affine group

Remarks

Example: on $\R^2$, the map $f(x) = Ax + b$ with $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $b = (1, 0)$ is in $\operatorname{AGL}(2, \R)$, since $\det A = 1 \ne 0$; its inverse is $x \mapsto A^{-1}(x - b)$. Non-example: $f(x) = Ax + b$ with $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ is affine but not bijective, so it is not in the affine group. Each 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. The map to the linear part $A$ is a surjective homomorphism onto the general linear group, whose kernel is the group of translations, isomorphic to $F^n$. Hence $\operatorname{AGL}(n, F) \cong F^n \rtimes \operatorname{GL}(n, F)$, a semidirect product. It is isomorphic to the group of $(n+1) \times (n+1)$ matrices $\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}$ with $A \in \operatorname{GL}(n, F)$.