For subsets $U,W,W'$ of $V$ is $U \oplus W = U \oplus W' \Rightarrow W=W'$ true? I have the following math homework quesion. I have found a solution saying this is false. However I can't seem to find an example or a reason for this to be false. 
The question is:
For subsets $U,W,W'$ of $V$ if $U \oplus W = U \oplus W'$ then is $W=W'$?
Thanks for any help.
 A: No and in $\Bbb R^2$ we have a simple counterexample:
$$\Bbb R^2=\operatorname{span}((1,0))\oplus\operatorname{span}((0,1))=\operatorname{span}((1,0))\oplus\operatorname{span}((1,1))$$
and 
$$\operatorname{span}((0,1))\ne\operatorname{span}((1,1))$$
A: Think of a plane in $\mathbb R^3$ that goes through the origin. Call it $U$
Consider two lines that go through the origin, that are not in the plane : $W$ and $W'$
Now $U \oplus W = U \oplus W' = \mathbb R^3$
But $W$ and $W'$ are distinct.
A: One very bizarre counterexample is to look at $\mathbb{R}$ and $\mathbb{R}^2$ as vector space over $\mathbb{Q}$ (not over $\mathbb{R}$ like we are used to). Note the following theorem:

Theorem (See Exercise 11.1.10-14 on p.414 of Dummit and Foote's Abstract Algebra) As vector spaces over $\mathbb{Q}$, $\mathbb{R}^n \cong \mathbb{R}$ for all $n \in \mathbb{Z}^+$. (Note that this also implies that they are isomorphic as additive groups.)

This shows that $\mathbb{R} \oplus \mathbb{R} \cong \mathbb{R} \oplus \{0\}$, but clearly $\mathbb{R} \not \cong \{0\}$.
