Vector space/Linear subspace/Translation/Equivalence relation/Fact

Let ${}K$ be a field, let ${}V$ be a ${}K$-vector space, and let ${}U\subseteq V$ denote a linear subspace.

Then the

relation given by

$v\sim w,{\text{ if }}v-w\in U,$

is an equivalence relation

on ${}V$.

Proof, Write another proof