Linear mapping/Eigenvalue zero/Characterization/Fact/Proof/Exercise

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

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

a linear mapping. Show that

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