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Let $V$ be a vector space of dimension $n$ over $F_q$. Prove that no of the subspace of dimension $k(\leq n)$ is equal to no of the subspace of dimension $n-k$ in $V$.

mahak batra
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1 Answers1

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The numbers are given by the Gaussian binomial coefficient $\binom nk_q$ and $\binom{n-k}{k}_q$. Both coincide.

Reference: How many k-dimensional subspaces there are in n-dimensional vector space over $\mathbb F_p$?

Dietrich Burde
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  • I think there is a more straightforward (?) argument using the bijection between the $k$-dimensional subspaces of $V$ and $n-k$-dimensional subspaces of $V^{*}$. (And there's a small typo in your answer, I reckon.) – cqfd Apr 16 '20 at 15:15