We consider an endomorphism $f$ of a $\mathbb{K}$-vector space $E$ of dimension $n \geqslant 2$ such that $f^n = 0$ and $f^{n-1} \neq 0$. III.B.1) Determine the set of vectors $u$ of $E$ such that the family $\mathcal{B}_{f,u} = (f^{n-i}(u))_{1 \leqslant i \leqslant n}$ is a basis of $E$. III.B.2) In the case where $\mathcal{B}_{f,u}$ is a basis of $E$, what is the matrix of $f$ in $\mathcal{B}_{f,u}$? III.B.3) Determine a basis of $E$ such that the matrix of $f$ in this basis is $A_{n-1}$. III.B.4) Give all subspaces of $E$ stable by $f$. How many are there? Give a simple relation between these stable subspaces and the kernels $\ker(f^i)$ for $i$ in $\llbracket 0, n \rrbracket$.
We consider an endomorphism $f$ of a $\mathbb{K}$-vector space $E$ of dimension $n \geqslant 2$ such that $f^n = 0$ and $f^{n-1} \neq 0$.
\textbf{III.B.1)} Determine the set of vectors $u$ of $E$ such that the family $\mathcal{B}_{f,u} = (f^{n-i}(u))_{1 \leqslant i \leqslant n}$ is a basis of $E$.
\textbf{III.B.2)} In the case where $\mathcal{B}_{f,u}$ is a basis of $E$, what is the matrix of $f$ in $\mathcal{B}_{f,u}$?
\textbf{III.B.3)} Determine a basis of $E$ such that the matrix of $f$ in this basis is $A_{n-1}$.
\textbf{III.B.4)} Give all subspaces of $E$ stable by $f$. How many are there? Give a simple relation between these stable subspaces and the kernels $\ker(f^i)$ for $i$ in $\llbracket 0, n \rrbracket$.