I was working on this homework problem:
Prove or give a counterexample: if $U_1, U_2, W$ are subspaces of $V$ such that $$ V = U_1 \oplus W \quad \text{and} \quad V = U_2 \oplus W$$ then $U_1 = U_2$.
One counterexample is: $$ V = \mathbb{F}^2, \quad W = \{(z,z) \mid z \in \mathbb{F} \}, \quad U_{1} = \{ (x,0) \mid x \in \mathbb{F} \} , \quad U_{2} = \{ (0,y) \mid y \in \mathbb{F} \} $$
I was wondering, for this given example, why is it true that $U_{1} \oplus W = \mathbb{F}^{2}$? Can anyone provide an explanation or proof of this claim?
$(x,y)=(x-y,0)+(y,y)$ so every vector is in the sum of $U_1$ and $W$. Of course, $U_1 \cap W=\{0\}$.