Vector space/Finite dimensional/Injective surjective bijective/Fact

Let ${}K$ denote a field, and let ${}V$ and ${}W$ denote ${}K$-vector spaces with the same dimension ${}n$. Let

$\varphi \colon V\longrightarrow W$

denote a linear mapping.

Then ${}\varphi$ is

injective if and only if ${}\varphi$ is

surjective.

Proof, Write another proof