# Direct Sum Uniqueness - Linear Algebra

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?

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Hint: No, it is not true. You can find a counter example by looking at a $2$-dimensional vector space with suitably chosen $1$-dimensional subspaces. –  Tobias Kildetoft Feb 4 '13 at 18:29
Why do you think thing you can cancel like that? It doesn't work. –  Thomas Andrews Feb 4 '13 at 18:36

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