Vector space and number of subspaces

[Buri]

Homework Statement

How many two dimensional subspaces does (F_3)^4 have?

The attempt at a solution

I chose an arbitrary basis so B = (v1,v2,v3,v4) for (F_3)^4 and then basically did 4C2 = 6 so it has 6 subspaces with dimension 2. However, thinking over this problem I've realized that I'm not exactly sure whether a different basis C can possibly generate another subspace that B generated. See if C were to generate another subspace that B didn't, wouldn't this mean that there is a vector in this subspace that can't be written as a linear combination of the vectors in B? And therefore, contradicting the fact that B is actually a basis for (F_3)^4?

This was a bonus question on my term test I just finished writing, so a general idea of how to arrive to a correct solution would be great.

Thanks!

EDIT: Another attempt at a solution

I've tried something else. Let U be subspace generated by two basis elements of B. Now I'd like to show that U is also generated by exactly two basis elements of C. So if I let x be an arbitrary vector in U we could write it as a linear combination of two basis elements of B:

x = a1v1 + a2v2 where a1 and a2 cannot both be zero.

So I would like to show that x = b1u1 + b2u2 where again b1 and b2 cannot both be zero.

Now u1 and u2 can be written as a linear linear combination of the vectors in B since they are in the vector space, but then I'd have to show that the some of the coefficients in this linear combination have to be zero (otherwise x would be a linear combination of 3 or 4 vectors of B contradicting that it was generated by only two vectors of B). But I can't get any where with this. Maybe the assertion isn't even true?

 

[lanedance]

apologies if its obvious, but what is $F_3$?

 

[Buri]

Its the finite field with elements {0,1,2} and arithmetic is performed modulo 3.

 

[Buri]

Any ideas?

 

[Hurkyl]

Try repeating your argument for a simpler problem, e.g. how many one-dimensional subspaces does F₃⁴ have? Or better, how about F₃²? Or maybe what one-dimensional subspaces does R² have?