Thread: Fields/Linear Algebra

I have a few questions:

1. Z_13=F and V_F=span((1,1),(1,2)) is (2,1) and (1,2) linearly independent in V_F?

I set up this system of linear equations, a(1,1)+b(1,2)=(2,1) and a(1,1)+b(1,2)=(1,2), the results are confusing to me, maybe im doing this wrong. For a(1,1)+b(1,2)=(2,1), I get a=3 b=-1 but does this say (2,1) is independent? For a(1,1)+b(1,2)=(1,2), I get a=1 b=0 obviously b(1,2)=(1,2) for b=1. so is (1,2) dependent?

2. Let V_Q be the set of all functions of a rational variable which are real valued. Is (x-3)/(x+sqrt(2)) and element of V_Q?

For this problem I know the -sqrt(2) is not a rational number so i cannot use that to get a real value. What else could I use?

3. |V_F|=49 implies 7^2, p=7. so the field is Z_7. How can I describe the collection of subspaces for V_F?

What if i tried solving a(1,1)+b(2,1)=(0,0) and a(1,1)+b(1,2)=(0,0). i then get a linearly independent spanning set. is that right?