In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra:
$$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{11}$ ▷ Using part 2, show that $\mathcal{A}_G$ is a vector subspace of $\mathcal{M}_n(\mathbf{R})$.