# Complement space of a finite dimensional space over a finite field

Let $$V$$ be a finite dimensional space over the field $$\mathbb{F}_q$$ of $$q$$ elements and let $$U\subset V$$ a subspace of $$V$$. How many subspaces $$W\subset V$$ are there such that $$W\cap U = 0$$ and $$V=W+U$$ ?

I've been trying to use Jordan canonical form but I think I'm missing something here, I just can't get it :(

Hint: count the number of ways that a given basis of $$U$$ can be completed into a basis of $$V$$, and divide by the number of bases of any fitting $$W$$.