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How many vectors in span({v})

by brru25
Tags: spanv, vectors
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brru25
#1
Sep16-09, 10:57 PM
P: 29
1. The problem statement, all variables and given/known data

Let v be an element of F2p \ {(0,0)}. How many vectors does Span({v}) have? How many 1-dimensional vector subspaces does F2p have?

F2p is the two-dimensional field (a,b) where each a, b are elements of Fp, where p is a prime number.

3. The attempt at a solution

I know the total number of elements in the vector space F2p \ {(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.
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution
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Office_Shredder
#2
Sep16-09, 11:02 PM
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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
#3
Sep16-09, 11:07 PM
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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
#4
Sep16-09, 11:17 PM
P: 29
How many vectors in span({v})

Quote Quote by Dick View Post
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
#5
Sep16-09, 11:21 PM
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Quote Quote by brru25 View Post
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
#6
Sep16-09, 11:22 PM
P: 29
Quote Quote by Dick View Post
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
#7
Sep16-09, 11:59 PM
P: 29
Quote Quote by Office_Shredder View Post
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.


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