Let $u \in \mathcal{L}(E)$. We are given two vectors $x$ and $y$ of $E$, as well as two integers $p \geq q \geq 1$ such that $u^{p}(x) = u^{q}(y) = 0$ and $u^{p-1}(x) \neq 0$. Show that the family $(x, u(x), \ldots, u^{p-1}(x))$ is free, and that if $(u^{p-1}(x), u^{q-1}(y))$ is free then $(x, u(x), \ldots, u^{p-1}(x), y, u(y), \ldots, u^{q-1}(y))$ is free.