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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 :(

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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 $W$.

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