Exercise 3, § 5

Let $u$ be an endomorphism of a finite dimensional vector space $E$, let $\lambda$ be an eigenvalue of $u$, and let $v$ be the endomorphism $u - \lambda$. Show that $E$ is the direct sum of $v^{-1}(0)$ and $v(E)$ if and only if $A$ is a simple root of the minimal polynomial of $u$.