Suppose that $u$ is an endomorphism of a vector space $E$ and $\dim({E})=n \ge2$. And supposing that $E$ is the only sub-space not equal to zero, and stable by $u$.
How to prove that:
$\forall x \in E\setminus\{0_E\}, (x,u(x),...,u^{n-1}(x))$ is a basis of $E$ ?
What is the form of the matrice of $u$ in this basis?
Hint
Let $F=\operatorname{span}(x,u(x),\ldots, u^{n-1}(x))$. Prove that $F$ is stable by $u$ so conclude that $F=E$. The family $(x,u(x),\ldots, u^{n-1}(x))$ with $n$ vectors generates $E$ so it's a basis for $E$.