Endomorphism/Invariant linear subspace/Polynomial/Action/Fact

Let ${}K$ be a field, ${}V$ a ${}K$-vector space and

$\varphi \colon V\longrightarrow V$

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

Then, for every polynomial

${}P\in K[X]$, the relation

${}P{\left(\varphi {|}_{U}\right)}={\left(P(\varphi )\right)}{|}_{U}\,$

holds, where here ${}\varphi {|}_{U}$ denotes the restricted mapping

(with respect to range and target).

Proof, Write another proof