Linear mapping/Kernel/Injectivity/Fact

Let ${\displaystyle {}K}$ denote a field, let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ denote ${\displaystyle {}K}$-vector spaces, and let

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

denote a ${\displaystyle {}K}$-linear mapping.

Then ${\displaystyle {}\varphi }$ is

injective if and only if ${\displaystyle {}\operatorname {kern} \varphi =0}$

holds.