Find the possible values of $\dim( U \cap W)$

Question

Here's the problem:

Let $V$ be a vector space and $U,W \subseteq V$. Assume $\dim(V)=7 ,\dim (U) = 4 $ and $\dim(W)= 5$. Find the possible values of $\dim( U \cap W)$.

I get $\dim(U \cap W) \in \{2,3,4\}$, by the dimension formula. Is this correct?

Answer 1

I'm not sure what exactly what you mean by the "dimension formula". That being said, we have $$ \dim(U) + \dim(W) = \dim(U \cap W) + \dim(U + W). $$ The left hand side must be $9$. Since $\dim(U + W) \in \{5,6,7\}$, we can conclude that $\dim(U \cap W) = 9 - \dim(U + W) \in \{2,3,4\}$.