Abstract
Let V be an n-dimensional vector space over the finite field F q 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
for the number of 1-dimensional subspaces of V. This gives us a clue of how to determine the general formula.
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© 2009 Hindustan Book Agency
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Cioabă, S.M., Murty, M.R. (2009). Block Designs. In: A First Course in Graph Theory and Combinatorics. Texts and Readings in Mathematics, vol 55. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-39-2_9
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DOI: https://doi.org/10.1007/978-93-86279-39-2_9
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-98-2
Online ISBN: 978-93-86279-39-2
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