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 $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$. Let $k \in \mathbf{N}^{*}$. Show that there exists a unique family $(f_{0}^{(k)}, \ldots, f_{k}^{(k)})$ of endomorphisms of $E$ such that $$\forall t \in \mathbf{R}, (u + tv)^{k} = \sum_{i=0}^{k} t^{i} f_{i}^{(k)}$$ Show in particular that $f_{0}^{(k)} = u^{k}$ and $f_{1}^{(k)} = \sum_{i=0}^{k-1} u^{i} v u^{k-1-i}$.
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 $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Let $k \in \mathbf{N}^{*}$. Show that there exists a unique family $(f_{0}^{(k)}, \ldots, f_{k}^{(k)})$ of endomorphisms of $E$ such that
$$\forall t \in \mathbf{R}, (u + tv)^{k} = \sum_{i=0}^{k} t^{i} f_{i}^{(k)}$$
Show in particular that $f_{0}^{(k)} = u^{k}$ and $f_{1}^{(k)} = \sum_{i=0}^{k-1} u^{i} v u^{k-1-i}$.