To clarify my question:
Let $W_1$ and $W_2$ be subspaces of a finite dimensional vector space $V$. What are the conditions that need to be satisfied in order to say that $\dim(W_1\cap W_2)=\dim(W_1)$?
I started by saying that if $W_1 \subseteq W_2$, then $W_1 \cap W_2=W_1$, in which case that $\dim(W_1 \cap W_2)$ must equal to $\dim(W_1)$. I also added the condition that the intersection of $W_1$ and $W_2$ can not be empty.
I'm having trouble trying to find out the conditions for other situations, like $W_1 \not\subset W_2$ and $W_2 \not\subset W_1$. I put down that if the basis of $W_1 \cap W_2$ has the order of $n$, then the basis of $W_1$ must have the same order. I'm not sure if I've done the right reasoning.
Any suggestion would help! Thanks a lot.