In this part, $\mathbf{K} = \mathbf{R}$. For every natural integer $n$, $n \geq 2$, we introduce the set, called the special linear group: $$\mathrm{SL}_n(\mathbf{R}) = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \operatorname{det}(M) = 1 \right\}.$$ If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ $\mathbf{9}$ ▷ Determine $\mathcal{A}_G$ when $G = \mathrm{SL}_n(\mathbf{R})$.
In this part, $\mathbf{K} = \mathbf{R}$. For every natural integer $n$, $n \geq 2$, we introduce the set, called the special linear group:
$$\mathrm{SL}_n(\mathbf{R}) = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \operatorname{det}(M) = 1 \right\}.$$
If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra:
$$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{9}$ ▷ Determine $\mathcal{A}_G$ when $G = \mathrm{SL}_n(\mathbf{R})$.