TITLE: Suppose $a,b$ are vectors in $V$ and $U,W$ are subspaces of $V$ such that $a + U = b + W$. Show that $U = W$.
QUESTION [1 upvotes]: Suppose  $a,b$ are vectors in $V$ and $U,W$ are subspaces of $V$ such that $a + U = b + W$. Show that  $U = W$.
attempt: We know that $a + U = {\{a + u : u \in U}\}$. And  $b + W = {\{b + w : w \in W}\}$. Then let $a= b + w$ for some $w \in W$. Then $w = a - b \in W$. 
And $b = a + u $ for some $u \in U$, so $u = b -a \in U$.
Then we have $a + u = a + (b-a) = b = b+w = b + (a-b) = a $. So $a = b$ for any $u \in U , w \in W.$
Can someone pleaese help me?  Thank you

REPLY [1 votes]: Here is the approach I think they are looking for: take a vector $w_0\in W$. Show, presumably by using the vectors $a,b,w$ and $u$, that $w_0\in U$ as well (the key observation here is that $a-b=w=-u$ is in both $U$ and $W$ simultaneously). In other words, show that $W\subseteq U$.
Then do it the other way around.
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