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[Let V be an n-dimensional vector space over the finite field Fq of q elements. We would like to determine the number of subspaces of dimension k. For example, the number of 1-dimensional subspaces is easily found as these are subspaces spanned by one element. Such an element must be non-zero and there are qn − 1 ways of choosing such an element. But for each choice, any non-zero scalar multiple of it will generate the same subspace as there are q − 1 such multiples for any fixed vector, we get a final tally of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{{q^{n - 1}}}}{{q - 1}}$$\end{document} for the number of 1-dimensional subspaces of V. This gives us a clue of how to determine the general formula.]
Published: May 24, 2017
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