Splitting of a maximal cyclic subspace Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ and let $u$ be an endomorphism of $V$. We assume that $u$ is nilpotent of index $n$, that is $u^n = 0$ and $u^{n-1} \neq 0$. We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero. a) Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis. b) Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$. c) Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.
\textbf{Splitting of a maximal cyclic subspace}\\
Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ and let $u$ be an endomorphism of $V$. We assume that $u$ is nilpotent of index $n$, that is $u^n = 0$ and $u^{n-1} \neq 0$. We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero.
a) Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis.
b) Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$.
c) Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.