Linear mapping/Kernel/Injectivity/Fact

Kernel criterion for injectivity

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

$\varphi \colon V\longrightarrow W$

denote a ${}K$-linear mapping.

Then ${}\varphi$ is

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

holds.

Proof, Write another proof