Show that the group of invertible affine transformations, $A(k^n)$, is a linear group, isomorphic to the subgroup of $GL_{n+1}(k)$ consisting of elements of the form $\begin{pmatrix}g & v \\ 0 & 1 \end{pmatrix}$, where $g \in GL_n(k)$ and $v \in k^n$.
I've shown that $A(k^n)$ is a group, but I'm unsure of how to construct the isomorphism required to show it is a linear group.
Any help would be greatly appreciated!
Let $G$ be your group. Conside the map $f\colon G\longrightarrow A(k^n)$ thus defined: if $\left(\begin{smallmatrix}g&v\\0&1\end{smallmatrix}\right)$, and if $w\in k^n$, then $f(g)(w)=gw+v$. Now, take $\left(\begin{smallmatrix}g'&v'\\0&1\end{smallmatrix}\right)\in G$. Then\begin{align}f\left(\begin{pmatrix}g&v\\0&1\end{pmatrix}.\begin{pmatrix}g'&v'\\0&1\end{pmatrix}\right)(w)&=f\begin{pmatrix}gg'&gv'+v\\0&1\end{pmatrix}(w)\\&=(gg')w+gv'+v\\&=g(g'w+v')+v\\&=f\begin{pmatrix}g&v\\0&1\end{pmatrix}\left(f\begin{pmatrix}g'&v'\\0&1\end{pmatrix}(w)\right).\end{align}Therefore, $f$ is a group homomorphism. Can you take it from here?