We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$. For every non-zero vector $x$ in $E$, we denote by $C_u(x)$ the vector space spanned by the $\left(u^k(x)\right)_{k \in \mathbb{N}}$ and $s(x)$ the smallest integer $\geqslant 1$ such that $u^{s(x)}(x) = 0$.
Prove that $(x, u(x), \ldots, u^{s(x)-1}(x))$ is a basis of $C_u(x)$ and give the matrix, in this basis, of the endomorphism induced by $u$ on $C_u(x)$.