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.
Hint: You may have $S\neq T$ but $S,T\subseteq R$.
No. Think about the following example. V is the plane, R is a line which passes the origin, S, T are different lines which also pass the origin.
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 $R+S=V$.
Another kind of example in infinite dimension:
Take $V=\mathbb R[X]$ and
$R=\mathbb R[X]_{\leq 6}$
$S=\mathbb R[X]_{\leq 3}$
$T=\mathbb R[X]_{\leq 2}$.
Then $R+S=R+T$ but $S\ne T$.