Let's say $U, W_1, W_2$ are subspaces of vector space $V$. Determine if following statements are true:
a) $U+W_1=U+W_2 \Rightarrow W_1=W_2$
b) $U \oplus W_1=V=U \oplus W_2 \Rightarrow W_1=W_2$
To be quite honest, i don't really know how could i prove this mathematically, anyway, i can see that in the case b) it is very likely that it is true because we have a direct sum, meaning that every vector in that sum can be represented as a sum of vectors from the $U$ and $W_1$ and that representation is UNIQUE, same thing works for $U$ and $W_2$ but that still ain't proof. Any help appreciated!
For $V=\Bbb{R}^2$, take $U$ as $x$-axis, $W_1$ as $y$ axis and $W_2=\text{span}\;(1,1)$ and see what will happen!