Linear mapping/Eigenvalue zero/Characterization/Fact

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

${\displaystyle \varphi \colon V\longrightarrow V}$

a linear mapping.

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