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 $\mathcal{A}^{\perp}$. Show that for every matrix $N \in \mathcal{A}$ and all $i \in \llbracket 1, r \rrbracket$, we have $N^{\top} A_{i} \in \mathcal{A}^{\perp}$.
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 $\mathcal{A}^{\perp}$.
Show that for every matrix $N \in \mathcal{A}$ and all $i \in \llbracket 1, r \rrbracket$, we have $N^{\top} A_{i} \in \mathcal{A}^{\perp}$.