Endomorphism/Eigenvalue and characteristic polynomial/Fact

Theorem about eigenvalues and characteristic polynomial

Let ${}K$ denote a field, and let ${}V$ denote an ${}n$-dimensional vector space. Let

$\varphi \colon V\longrightarrow V$

denote a linear mapping.

Then

${}\lambda \in K$ is an eigenvalue of ${}\varphi$ if and only if ${}\lambda$ is a zero of the characteristic polynomial

${}\chi _{\varphi }$.

Proof, Write another proof