Dimension proof of the intersection of 3 subspaces

[harvesl]

Homework Statement

Assume V = \mathbb{R}^n where n \geq 3. Suppose that U,W,X are three distinct subspaces of dimension n-1; is it true then that dim(U \cap W \cap X) = n-3? Either give a proof, or find a counterexample.

The Attempt at a Solution

The question previous to this was showing that for subspaces U,W of dimension n-1 \longrightarrow dim(U \cap W) = n - 2 which I was able to prove fine. Now, I'm thinking this is true, because W \cap X will be a subspace of dimension n-2 so if we set W \cap X = Z, where Z is a subspace we turn this problem into dim(U \cap Z) and since all three were distinct we should have dim(U \cap Z) = n-3 but I'm not entirely sure.

Thanks!

 

[voko]

You seem to be on the right track.

 

[harvesl]

You seem to be on the right track.

Thanks.