In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ We recall that, if $M$ is a matrix in $\mathcal{M}_n(\mathbf{R})$, we say that $M$ is tangent to $G$ at $I_n$ if there exist $\varepsilon > 0$ and an application $\left.\gamma : \right]-\varepsilon, \varepsilon[ \rightarrow G$, differentiable, such that $\gamma(0) = I_n$ and $\gamma'(0) = M$. The set of matrices tangent to $G$ at $I_n$ is called the tangent space to $G$ at $I_n$, and is denoted $\mathcal{T}_{I_n}(G)$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$. $\mathbf{14}$ ▷ Prove the inclusion $\mathcal{A}_G \subset \mathcal{T}_{I_n}(G)$.
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra:
$$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
We recall that, if $M$ is a matrix in $\mathcal{M}_n(\mathbf{R})$, we say that $M$ is tangent to $G$ at $I_n$ if there exist $\varepsilon > 0$ and an application $\left.\gamma : \right]-\varepsilon, \varepsilon[ \rightarrow G$, differentiable, such that $\gamma(0) = I_n$ and $\gamma'(0) = M$. The set of matrices tangent to $G$ at $I_n$ is called the tangent space to $G$ at $I_n$, and is denoted $\mathcal{T}_{I_n}(G)$.
In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{14}$ ▷ Prove the inclusion $\mathcal{A}_G \subset \mathcal{T}_{I_n}(G)$.