Endomorphism/Finite-dimensional/Minimal polynomial/Principal ideal/Fact

Let ${}V$ be a finite-dimensional vector space over a field ${}K$, and let

$f\colon V\longrightarrow V$

denote a linear mapping.

Then the set

${\left\{P\in K[X]\mid P(f)=0\right\}}$

is a principal ideal in the polynomial ring ${}K[X]$, which is generated by the minimal polynomial

${}\mu _{f}$.

Write a proof