Just randomly making up some subspaces

[roeb]

Homework Statement

Prove or give a counterexample: if U₁, U₂, W are subspaces of V such that:

U₁ + W = U₂ + W then U₁ = U₂

Homework Equations

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?

 

[Office_Shredder]

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

 

[HallsofIvy]

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?