Vector spaces/Linear mapping/Factorization/Fact

Factorization theorem (vector spaces)

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

$\varphi \colon V\longrightarrow W$

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

Then there exists a canonical factorization

$V{\stackrel {q}{\longrightarrow }}V/\operatorname {kern} \varphi {\stackrel {\theta }{\longrightarrow }}\operatorname {Im} \varphi {\stackrel {\iota }{\hookrightarrow }}W,$

where ${}q$ is the canonical projection, ${}\theta$ is a vector space isomorphism, and ${}i$ is the canonical inclusion of the image space

in ${}W$.

Proof, Write another proof