Lie algebra of affine linear maps

Question

Let $G$ be the Lie group of affine transformations, $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ We can represent these maps as matrices $$\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}.$$

Now, I read that one can describe the Lie algebra $\mathfrak{g}$ as the set of matrices $\begin{pmatrix} A & b \\ 0 & 0 \end{pmatrix}$ How can I rigorously show that this is indeed the Lie algebra of $G$?

Answer 1

Hint The elements of the Lie algebra $\mathfrak{g} \cong T_{\Bbb I} G$ are the tangent vectors at the identity element $\Bbb I \in G$ to curves in $G$ through that point.