# Dimension proof of the intersection of 3 subspaces

1. Jan 13, 2013

### harvesl

1. The problem statement, all variables and given/known data

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.

3. 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!

2. Jan 13, 2013

### voko

You seem to be on the right track.

3. Jan 13, 2013

Thanks.