We introduce the area of extremal set theory via three classical results: the Erdos-Ko Rado theorem, Frankl's $r$-wise intersection theorem, and the Kruskal-Katona shadow theorem. We then consider vector space analogs of these problems. We prove a vector space analog of a version of the Kruskal-Katona theorem due to Lov\'{a}sz. We apply this result to extend Frank's theorem on $r$-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erdos-Ko-Rado theorem for vector spaces.