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Prove or provide a counterexample:

Let $V$ be a vector space, $U_1, U_2$ subspaces of $V$. If there exists a subspace $W \subseteq V$ such that $$U_1\oplus W=U_2\oplus W,$$ then $U_1=U_2$.

I can easily come up with a counterexample for the statement if those are simply sums instead of direct sums. (Something like $U_1=0, U_2=V, W=V$, then $U_1+W=U_2+W=V$ but $U_1\neq U_2$.) But I can't think of any counterexample for direct sums, and now I'm left wondering whether this statement is true or not.

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    $\begingroup$ Take $V=\mathbb R^2$, $U_1=\mathrm{span}(1,0)$, $U_2=\mathrm{span}(1,1)$ and $W=\mathrm{span}(0,1)$. Then $U_1\oplus W= U_2\oplus W=V$, but $U_1\neq U_2$. $\endgroup$
    – SMM
    Commented Jan 15, 2015 at 23:27
  • $\begingroup$ Note that $\dim (A \oplus B) = \dim A + \dim B$, so if $V$ is finite any counterexample must satisfy $\dim U_1 = \dim U_2$. $\endgroup$
    – Travis Willse
    Commented Jan 15, 2015 at 23:33

1 Answer 1

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Any two distinct lines through the origin ($1$-dimensional subspaces) span $\mathbb{R}^2$.

So, pick two distinct lines $U_1, U_2$ through the origin in $\mathbb{R}^2$. Can you pick another line $W$ through the origin such that $U_1$ and $W$ span $\mathbb{R}^2$ and so do $U_2$ and $W$?

Remark Note that this counterexample is minimal in the sense that the statement is true when $\dim V \leq 1$.

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