Exercise 1, § 5

Let $u$ be an endomorphism of a finite dimensional vector space $E$, and let $V$ be a subspace of $E$ invariant under $u$. Show that the characteristic polynomial of the restriction of $u$ to $V$ divides that of $u$. Hence obtain an elementary proof of the Cayley-Hamilton theorem (reduce to the case where $E_u$ is cyclic, and use formula (2) of VII, p. 30).