# How to prove this affine subspace? [closed]

Let $$V$$ be a vector space and $$W$$ a subspace of $$V$$. Let an equivalence relation on V be given by $$x R y$$ iff $$x-y \in W$$. I need to show that the equivalence class of some vector $$x$$ in $$V$$ forms an affine subspace of $$V$$, defined as the translated subspace $$W$$. I've tried some ideas but none get me to the result.

## closed as off-topic by RRL, Theoretical Economist, Leucippus, Feng Shao, Daniele TampieriSep 13 at 9:15

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The equivalence class of $$x$$ is nothing but $$x+W$$ which is affine.
Let $$v\in V$$. Then $$v+W=\{v+w\mid w\in W\}$$ is an equivalence class, since $$vRy$$ or by symmetry $$yRv$$ means that $$y-v\in W$$, that is $$y\in v+W$$.