So how does that use the contradiction we've established? Can you clarify?

I didn't do it by contradiction but directly......$\displaystyle \dim(U\cap W)>0$ is the same as $\displaystyle \dim(U\cap W)\geq 1$ as we're dealing with integer numbers. Tonio
Also, we haven't gotten that far in the class to show that equation you used, so I would have to show that too somehow.

I'll ask him when we get done with spring break, but something tells me that's not what the professor wants (i.e. is there another way of doing it?).