Endomorphism/Invariant linear subspace/Minimal polynomial/Divisibility/Fact/Proof

Proof

Let ${}\mu$ be the minimal polynomial of ${}\varphi$. For ${}u\in U$, we have

${}\mu {\left(\varphi {|}_{U}\right)}(u)=\mu (\varphi )(u)=0\,,$

due to fact. Therefore, ${}\mu$ annihilates the restricted endomorphism ${}\varphi {|}_{U}$, and so ${}\mu$ is a multiple of the minimal polynomial of ${}\varphi {|}_{U}$.

To fact