Vector space/Linear subspace/Residue class space/Fact

Let ${}K$ be a field, let ${}V$ be a ${}K$-vector space, and let ${}U\subseteq V$ denote a linear subspace. Let ${}V/U$ be the set of all equivalence classes (the quotient set) of the equivalence relation on ${}V$ defined by ${}U$, and let

$q\colon V\longrightarrow V/U,v\longmapsto [v],$

denote the canonical projection.

Then there exists a uniquely determined structure of a ${}K$-vector space on ${}V/U$ such that ${}q$ is a ${}K$-linear mapping.

Proof, Write another proof