We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$$ There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$, $x \in L^{\perp}$, and $\mathcal{V} x \subset L^{\perp}$. Justify that $\lambda x \notin \mathcal{V} x$ for every $\lambda \in \mathbf{R}^{*}$, and deduce from the two previous questions that $$\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$$
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$$
There exists a vector subspace $L$ of $E$ such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$, $x \in L^{\perp}$, and $\mathcal{V} x \subset L^{\perp}$.
Justify that $\lambda x \notin \mathcal{V} x$ for every $\lambda \in \mathbf{R}^{*}$, and deduce from the two previous questions that
$$\operatorname{dim} \mathcal{V} x + \operatorname{dim} L \leq n-1$$