Vector spaces/Linear mapping/Homomorphism theorem/Surjective and kernel/Fact

Homomorphism theorem (vector spaces)

Let ${}K$ be a field, and let ${}V,Q,$ and ${}W$ denote vector spaces over ${}K$. Let ${}\varphi \colon V\rightarrow W$ be a linear mapping, and ${}\psi \colon V\rightarrow Q$ be a surjective linear mapping. Suppose that

${}\operatorname {kern} \psi \subseteq \operatorname {kern} \varphi \,$

holds.

Then there exists a uniquely determined linear mapping

${\tilde {\varphi }}\colon Q\longrightarrow W$

fulfilling

${}\varphi ={\tilde {\varphi }}\circ \psi$.

In other words: the diagram

${\begin{matrix}V&{\stackrel {\varphi }{\longrightarrow }}&W&\\\!\!\!\!\!\,\,\psi \downarrow &\nearrow {\tilde {\varphi }}\!\!\!\!\!&\\Q&&&&\!\!\!\!\!\!\!\!\\\end{matrix}}$

commutes.

Proof 1, 2, Write another proof