Linear mapping/Eigenvalue zero/Characterization/Fact

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

$\varphi \colon V\longrightarrow V$

a linear mapping.

Then

${}\operatorname {ker} {\left(\varphi \right)}=\operatorname {Eig} _{0}{\left(\varphi \right)}\,.$

In particular, ${}0$ is an eigenvalue of ${}\varphi$ if and only if ${}\varphi$ is not injective.

Write a proof