Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$. We denote by $\mathcal{M}_{n,1}(\mathbb{R})$ the $\mathbb{R}$-vector space of column matrices with $n$ rows and real coefficients. Let $X \in \mathcal{M}_{n,1}(\mathbb{R})$ and let $F = \operatorname{Vect}(A_{1}X, \ldots, A_{r}X)$. Show that $F$ is stable by the endomorphisms of $\mathcal{M}_{n,1}(\mathbb{R})$ canonically associated with elements of $\mathcal{A}^{\top}$.
Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$. We denote by $\mathcal{M}_{n,1}(\mathbb{R})$ the $\mathbb{R}$-vector space of column matrices with $n$ rows and real coefficients.
Let $X \in \mathcal{M}_{n,1}(\mathbb{R})$ and let $F = \operatorname{Vect}(A_{1}X, \ldots, A_{r}X)$. Show that $F$ is stable by the endomorphisms of $\mathcal{M}_{n,1}(\mathbb{R})$ canonically associated with elements of $\mathcal{A}^{\top}$.