I encountered this question in a document I found on a google search, it bugged me because my perception keeps telling me I'm wrong no matter what I do.
Let $U$, $W$ and $Z$ be subspaces of a vector space $V$ and the following conditions are satisfied:
- $U + Z = W +Z$
- $U \cap Z = W\cap Z$
- $U \subset W$
Prove that $U=W$
So I tried proving by contradiction that $U$ is not equal to $W$, by saying there exists some vector $w \in W \setminus U$ and if $w \in W \cap Z$ then $w \in U \cap Z$ which means $w \in U$. At this momment my mind is shouting "This isn't it", and I think I should be using a different approach.