Consider a finite field $K$ with $q$ elements. For each $0\leq k\leq n$, there are ${n \brack k}_q$ subspaces of $K^n$ of dimension $k$. I want to count cosets. Let $A$ be a subspace of $K^n$ of dimension $k$. By simple group theory, there are $[K^n:A]=q^n/|A|$ cosets of the form $x+A$ $(x\in K^n)$. I am not sure how to combine this with ${n \brack k}_q$ to get the number of cosets of dimension $k$. Any suggestions?
2026-05-05 11:43:54.1777981434
Counting cosets
109 Views Asked by user147263 https://math.techqa.club/user/user147263/detail AtRelated Questions in LINEAR-ALGEBRA
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