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

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Let ${}U\subseteq V$ be a ${}\varphi$-invariant linear subspace and

$\varphi {|}_{U}\colon U\longrightarrow U$

the restriction to ${}U$ (also in the target).

Then the

minimal polynomial

of ${}\varphi$ is a multiple of the minimal polynomial of ${}\varphi {|}_{U}$.

Proof, Write another proof