Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

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 $b \in GF(q)$.

share|improve this answer
    
Oh yes! it is $q^2$ rather than $q^n$. –  Babak 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? –  Babak 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. –  Babak S. May 7 '12 at 6:08

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.