Suppose that $V$ is vector space with dimension $p^2$ defined on a finite field $F$. How many subspaces of dimension one, $V$ has?
Find the number of subspaces
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linear-algebra
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1You mean $V$ is a vector space, I think, and I suppose $F$ is a finite field. – 2012-05-25
1 Answers
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Hint: each subspace of dimension $1$ is spanned by a nonzero vector. How many of those are there? How many vectors span the same subspace?
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0I know that we have at most $q^2-1$ elements. But I was told that there are $q-1$ elements reiteratively. I don't know how this later occurs. – 2012-05-25
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0How many elements does the field $F$ have? – 2012-05-25
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0I am so sorry. The Field has $p$ elements and I confused $p$ with $q$ in my previuos comment. – 2012-05-25
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1So there are $p^{p^2}$ vectors in $V$, of which all but $1$ are nonzero. Now, how many span the same $1$-dimensional subspace? – 2012-05-25