Problem
Prove or give counterexample: if $U_1$, $U_2$, $W$ are subspaces of $V$ such that $$U_1 + W = U_2 + W$$ then $U_1 = U_2$
Solution
Here’s a counterexample: let $W = \lbrace (x,y,0) | x,y \in \mathbb{F} \rbrace$, $U_1 = \lbrace (0,0,z) | z \in \mathbb{F} \rbrace$ and $U_2 = \lbrace (0,z,z) | z \in \mathbb{F} \rbrace$, then by example 1.43: $$U_1 + W \stackrel{D1.36}{=} \mathbb{F}^3$$ But we also have: $$U_2 + W \stackrel{D1.36}{=} \mathbb{F}^3$$ Therefore $U_1 + W = U_2 + W$, but clearly $U_1 \neq U_2$.