We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$).
Let $x$ and $y$ be two elements of $E$, with $x$ being non-zero. Show that there exists $u \in \mathcal{A}$ such that $u(x) = y$.
One may consider in $E$ the vector subspace $\{ u(x) \mid u \in \mathcal{A} \}$.