Problem
Prove or give counterexample: if $U_1$, $U_2$, $W$ are subspaces of $V$ such that $$U_1 \oplus W = U_2 \oplus W$$ then $U_1 = U_2$
Solution
We have that $U_1 \oplus W = V = U_2 \oplus W$, therefore $U_1 \cap W = U_2 \cap W = \lbrace 0 \rbrace$. Choosing any element $u \in U_1$ will therefore not be in $W$, but will be in $V$ as $U_1 \subseteq V$.
Since $u$ $\cancel{\in}$ $W$ and since $u \in V = U_2 \oplus W$ we must have that $u \in U_2$, and by a similar argument $u \in U_2$ implies $u \in U_1$. Therefore $U_1 \subseteq U_2$ and $U_1 \subseteq U_2$ and thus $U_1 = U_2$.