In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. There exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal, denoted $\langle u, v \rangle$ or $u \cdot v$. Let $H$ be a hyperplane of $E$ and $D$ its orthogonal complement. If $(u)$ is a basis of $D$ and if $U$ is the column matrix of $u$ in $\mathcal{B}$, show that $H$ is stable by $f$ if and only if $U$ is an eigenvector of the transpose of $A$.
In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. There exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal, denoted $\langle u, v \rangle$ or $u \cdot v$.
Let $H$ be a hyperplane of $E$ and $D$ its orthogonal complement. If $(u)$ is a basis of $D$ and if $U$ is the column matrix of $u$ in $\mathcal{B}$, show that $H$ is stable by $f$ if and only if $U$ is an eigenvector of the transpose of $A$.