1. The problem statement, all variables and given/known data Let v be an element of F^{2}_{p} \ {(0,0)}. How many vectors does Span({v}) have? How many 1-dimensional vector subspaces does F^{2}_{p} have? F^{2}_{p} is the two-dimensional field (a,b) where each a, b are elements of F_{p}, where p is a prime number. 3. The attempt at a solution I know the total number of elements in the vector space F^{2}_{p} \ {(0,0)} is p^{2} - 1. I also thought that this was the number of vectors in Span({v}) but I've told by a couple people that is not so. I started with the definition of span but I just couldn't see the rest unfold. Thank you ahead of time for your help. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
An element in span(v) must be of the form av for some a in the field. How many choices of a do you have? Do all of these choices yield a different vector, or can av=bv if a=/=b?
Take an example. Let F=Z_3, the field with three elements {0,1,2}. Yes, there are 8 elements in (Z_3)^2-{0,0}. How many elements are in span((1,1))? That's (1,1)*x for all x in Z_3. Does that help you to see things unfold?
Well, yeah. Would have three elements. But they would be (0,0)=(1,1)*0, (1,1)=(1,1)*1 and (2,2)=(1,1)*2.