Let $k$ be a field, let $U$, $V$, and $W$ be vector spaces over $k$, and let $i:U\to V$ and $q:V\to W$ be $k$-linear maps such that $i$ is injective, $\operatorname{Range}(i)=\ker(q)$, and $q$ is surjective. Then there exists a unique $k$-[linear map](/page/Linear%20Map)
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\begin{align*}
\Phi:V/\operatorname{Range}(i)\to W
\end{align*}