How Many Vectors Are in the Span of a Single Vector in F2p?

[brru25]

Homework Statement

Let v be an element of F²_p \ {(0,0)}. How many vectors does Span({v}) have? How many 1-dimensional vector subspaces does F²_p have?

F²_p is the two-dimensional field (a,b) where each a, b are elements of F_p, where p is a prime number.

The Attempt at a Solution

I know the total number of elements in the vector space F²_p \ {(0,0)} is p² - 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.

 

[Office_Shredder]

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?

 

[Dick]

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?

 

[brru25]

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?

Would (1,1)*x have 3 elements? (1,1,0), (1,1,1), (1,1,2)? Or am I reading this incorrectly?

 

[Dick]

Would (1,1)*x have 3 elements? (1,1,0), (1,1,1), (1,1,2)? Or am I reading this incorrectly?

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.

 

[brru25]

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.

Okay now I see what you meant sorry about that.

 

[brru25]

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?

"a" I believe can be any number in F.