Vector spaces/Linear mapping/Factorization/Fact

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

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

${\displaystyle {}W}$