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Suppose that $V=V_{2}(q)$ is a vector space on a finite field $GF(q)$, so $|V|=q^{2}$. I saw this problem somewhere, " Describe one dimension subspaces of $V$ and find the number of them".

What I have done for this problem:

"I know that if we take such subspace, it would be like $<v>= \{ av|a\in GF(q) \} $. So we have the number $(q^2-1)/(q-1)$ of one-dimension subspaces as required."

For the rest of above problem any help or hint will be appreciated. Thanks.

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1 Answer

up vote 1 down vote accepted

You mean $q^n$ rather than $q^2$. The idea is that $v$ can be any member of $V$ except $0$, and that $v$ and $bv$ span the same one-dimensional space for any nonzero $b \in GF(q)$.

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Oh yes! it is $q^2$ rather than $q^n$. –  B. S. May 6 '12 at 18:48
So, you meant forexample $<[0,1]>$ and $<[0,b]>$ could describe a one-dimention subspace as required above? –  B. S. May 6 '12 at 18:54
@Babak, you do know what 1-dimensional subspaces of the the real plane $\mathbf{R}^2$ look like, don't you? The vectors $\mathbf{j}$ and $b\mathbf{j}, b\neq0,$ generate the same subspace. The same business here. –  Jyrki Lahtonen May 7 '12 at 5:14
@JyrkiLahtonen: As you pointed, I knew that and no doubt in what Prof. Israel wrote here. But, I wanted to probe this problem in the group theory instead of linear algebra. In fact, I wanted to examine this problem with a proper action on a proper set. I just thought about it. Any way thank you for the comment. –  B. S. May 7 '12 at 6:08
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