Vector subspace equality proof/disproof. Given $R,S,T$ are subspaces of vector space $V$, and $R+S=R+T$, does it follow $S=T$?
Please don't give a full proof, but some general help would be much appreciated. I get the basic idea that to show $S=T$ would be to show them to be subsets of one another. Not sure how to do this in a concrete way though.

Your intuition is wrong, $R+S=R+T$ does not imply $S=T$.
Let $V=\mathbb{R}^2$ and $R=\mathbb{R}(1,0)$, then there exists an infinity of $S$ such that $R+S=V$.
