Question regarding subspaces and dimensions

Question

I have a hw question I am stuck on.

Two vectors spaces $X$ and $Y$ are two subspaces in $\mathbb{R}^n.$ If $\dim(X)$ + $\dim(Y) > n$, how can I prove that there must exist a non-zero vector in their intersection?

I know that the dimension of $\mathbb{R}^n$ is $n$, but this is where I'm kinda stuck. Any help or useful links would be appreciated. I've been googling to no resources to avail so far

As originally written, your question was missing context. I believe I have fixed the issue, but please clarify if this was not the correct edit. — JohnColtraneisJC, Sep 03 '19 at 17:31
Are you sure dim$(U)>n$ and dim$(V)>n$? Subspaces have dimension no greater than the ambient space. — postmortes, Sep 03 '19 at 17:32
Contrapositively, if $X \cap Y = {0}$, then $X + Y = X \oplus Y \subseteq \Bbb R^n$... — Travis Willse, Sep 03 '19 at 18:36
The proof will be based on the fact that they will intersect at the zero vector. — Phil H, Sep 03 '19 at 19:09

score 0 · Answer 1 · answered Sep 04 '19 at 00:17

Suppose not. That is, $\dim(X)+\dim(Y)>n \ $ but $\ X\cap Y = \{ 0\}$.

We have the formula (here is a discussion on this formula) $$\dim(X+Y)=\dim(X)+\dim(Y)-\dim(X\cap Y)$$

Then if the intersection is $\{0\}$ we have that $\dim(X+Y)=\dim(X)+\dim(Y)>n$, but $X+Y$ is in a vector space with dimension $n$; contradiction!

Question regarding subspaces and dimensions

1 Answers1