Exercise 3.4.45

If the left side of $\dim <!--\; FUNCTION\; APPLICATION\; -->(V)+\dim <!--\; FUNCTION\; APPLICATION\; -->(W)=\dim <!--\; FUNCTION\; APPLICATION\; -->(V\cap W)+\dim <!--\; FUNCTION\; APPLICATION\; -->(V+W)$ is greater than $n$, then $\dim <!--\; FUNCTION\; APPLICATION\; -->(V\cap W)$ must be greater than zero. So $V\cap W$ contains nonzero vectors. Oh here is a more basic approach: Put a basis for $V$ and then a basis for $W$ in the columns of a matrix $A$. Then $A$ has more columns than rows and there is a nonzero solution to $\mathit{Ax}=0$. That $x$ gives a combination of the $V$ columns $=$ a combination of the W columns.