Linear algebra: $V$ is the direct sum of $U_1$ and $W$ and of $U_2$ and $W$: are $U_1$ and $U_2$ equal?

Question

If $V = U_1 \oplus U_2$ and $V = U_2 \oplus W$ then $U_1 = U_2$?

$V$ is a complex vector space and $U_1$, $U_2$ and $W$ subspaces thereof. $\oplus$ denotes the direct sum. Please prove or give a counterexample.

Answer 1

False.

Take $V=\mathbb{C}^2$, $W=\text{Span}(0,1)$, $U_1=\text{Span}(1,0)$ and $U_2=\text{Span}(1,1)$.