We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, and the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$. Let $y \in E$. Prove that $f_{1}^{(p-1)}(y) \in K(\mathcal{V})$. Using a relation between $u(f_{1}^{(p-1)}(y))$ and $v(u^{p-1}(y))$, deduce that $v(x) \in u(K(\mathcal{V}))$ for every $x \in \operatorname{Im} u^{p-1}$.
We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, and the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Let $y \in E$. Prove that $f_{1}^{(p-1)}(y) \in K(\mathcal{V})$. Using a relation between $u(f_{1}^{(p-1)}(y))$ and $v(u^{p-1}(y))$, deduce that $v(x) \in u(K(\mathcal{V}))$ for every $x \in \operatorname{Im} u^{p-1}$.