0
$\begingroup$

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 $= \{ 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.

1 Answers 1

1

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 $b \in GF(q)$.

  • 0
    @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.2012-05-07