Dimension proof of the intersection of 3 subspaces

[harvesl]

Homework Statement

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

The Attempt at a Solution

The question previous to this was showing that for subspaces [itex]U,W[/itex] of dimension [itex]n-1 \longrightarrow dim(U \cap W) = n - 2[/itex] which I was able to prove fine. Now, I'm thinking this is true, because [itex]W \cap X[/itex] will be a subspace of dimension [itex]n-2[/itex] so if we set [itex]W \cap X = Z[/itex], where [itex]Z[/itex] is a subspace we turn this problem into [itex]dim(U \cap Z)[/itex] and since all three were distinct we should have [itex]dim(U \cap Z) = n-3[/itex] 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.