# Just randomly making up some subspaces

1. Sep 20, 2009

### roeb

1. The problem statement, all variables and given/known data
Prove or give a counterexample: if U1, U2, W are subspaces of V such that:

U1 + W = U2 + W then U1 = U2

2. Relevant equations

3. The attempt at a solution
I would be inclined to say that it's true, however I took a peek at the back of the book and that's incorrect. Here's why I thought it was correct.

U1 = {(x,y) ∈ F2: y = x}
W = {(x,y) ∈ F2: x ∈ F, y ∈ F}
(just randomly making up some subspaces)

U1 + W = W + something
then that something has to be U1 = U2

I can't think of any U1 that would change if I add W to it.

Does anyone have any suggestions?

2. Sep 20, 2009

### Office_Shredder

Staff Emeritus
Re: Subspaces

Ok, but you can't just pick U1 and W and demonstrate it's true for only that case. What if W=V?

3. Sep 20, 2009

### HallsofIvy

Staff Emeritus
Re: Subspaces

Suppose V= $R^2$, $U_1= \{k\vec{i}\}$ for k a real number, $U_2= R^2$ and W= $R^2$. What are U1+ W and U2+ W?