# Thread: Proving The Dimension Of Intersection Of Two Subspaces Is Greater Than Or Equal To 1

1. ## Proving The Dimension Of Intersection Of Two Subspaces Is Greater Than Or Equal To 1

Let V be a vector space of dimension n. Let $\displaystyle W \subset V$ be a subspace of dimension $\displaystyle 1 \le k \le n$. Let $\displaystyle U \subset V$ be a subspace of dimension $\displaystyle 1 \le m \le n$. Prove that if $\displaystyle m > n-k$, then:

$\displaystyle dim(U \cap W) \ge 1.$

The hint says to argue by contradiction. So we assume $\displaystyle dim(U \cap W) < 1$. So if that's the case, $\displaystyle dim(U \cap W) = 0$ since dimension is non-negative. It now says to use the concept of dimension to produce the contradiction. Here's where I'm getting stuck.

2. Originally Posted by mathematicalbagpiper

Let V be a vector space of dimension n. Let $\displaystyle W \subset V$ be a subspace of dimension $\displaystyle 1 \le k \le n$. Let $\displaystyle U \subset V$ be a subspace of dimension $\displaystyle 1 \le m \le n$. Prove that if $\displaystyle m > n-k$, then:

$\displaystyle dim(U \cap W) \ge 1.$

The hint says to argue by contradiction. So we assume $\displaystyle dim(U \cap W) < 1$. So if that's the case, $\displaystyle dim(U \cap W) = 0$ since dimension is non-negative. It now says to use the concept of dimension to produce the contradiction. Here's where I'm getting stuck.

In general, $\displaystyle \dim(U+W)=\dim U+\dim W-\dim(U\cap W)$ , so:

$\displaystyle n\geq \dim(U+W)=\dim U+\dim W-\dim(U\cap W)= m+k-\dim(U\cap W)$ $\displaystyle \Longrightarrow \dim(U\cap W)\geq m+k-n>0$ , the last inequality following from

the given data, and thus we're done.

Tonio

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

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?).

4. Originally Posted by mathematicalbagpiper
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?).

.