# Show that $E / \sim$ is an affine space over $V/U$

Let $E$ be an affine space over the vector space $V$, and let $U \subseteq V$ be a vector subspace. We define the equivalence relation $$P \sim Q := \exists v \in U \text{ such that } P = Q + v$$ on $E$.

I have to show that $E / \sim$ is an affine space over the quotient space $V/U$.

Therefore, I would simply check the properties of an affine space, for example $$P + 0 = P.$$ Since we work on $E / \sim$, and since we defined $\sim$ on $E$, its elements have to be equivalence classes. Therefore, I actually have to show something like $$[P] + 0 = [P].$$ Now I'm wondering which rules of addition I have to apply. We've already proven that on a quotient space such as $V/U$ I'm allowed to write something like $$[v] + [w] = [v + w].$$ Is there something similar for the addition on affine spaces?

• You should indeed define an addition of vectors of $V/U$ on $E/\sim$ in order to endow it with an affine space structure. $[P]+[v]=[P+v]$ seems like a nice idea, don't you think? – Arnaud D. Jun 5 '16 at 13:37
• Thanks for editing my formatting! Let me take a guess: In order to prove the statement above, I have to show that $$[P + v] = [P' + v']$$ for $$[P] = [P']$$ and $$[v] = [v'].$$ – Julian Jun 5 '16 at 13:47
• Yes, you need to do that to prove that the addition I suggested is well-defined. Once this is done you can try proving that the axioms of affince spaces hold. – Arnaud D. Jun 5 '16 at 14:04
• And I guess I already figured it out. Assume $$[P] = [P'] ^ [v] = [v']$$. Then $$P' \in [P] => P' ~ P <=> \exists u \in U: P' = P + u (and thus \exists u' \in U: v' = v + u').$$ This leads to $$P' + v' = P + v + u + u'$$ and since $$u + u' \in U$$ it follows that $$P' + v' ~ P + v => [P' + v'] = [P + v].$$ – Julian Jun 5 '16 at 14:08
• Yes, that's correct :) The rest shouldn't be very hard. – Arnaud D. Jun 5 '16 at 14:09