Vector spaces/Linear mapping/Surjective and residue class space/Fact

Isomorphs theorem (vector spaces)

Let ${}K$ be a field, and let

$\varphi \colon V\longrightarrow W$

denote a surjective linear mapping between ${}K$-vector spaces.

Then there exists a canonical

linear isomorphism

${\tilde {\varphi }}\colon V/\operatorname {kern} \varphi \longrightarrow W.$

Proof, Write another proof