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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.

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

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The equivalence class of $x$ is nothing but $x+W$ which is affine.

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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 $y\in v+W$.

All equivalence classes have this form.

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