Direct Sum Uniqueness - Linear Algebra

Question

Prove or disprove: If $U_1, U_2, W$ are subspaces of a vector space $V$ satisfying $U_1 \bigoplus W = U_2 \bigoplus W$, then $U_1=U_2$.

My first thought is that this statement is true. Since $U_1 \bigoplus W = U_2 \bigoplus W$, then $U_1+W=U_2+W$ which means $U_1=U_2$. Is that all I need to do?

Answer 1

Nope. Take $V = \mathbb{R}^2$, say, and any three distinct one-dimensional subspaces $U_1, U_2, W$.