Hi, I have to find the number of subspaces of where is a finite field with elements.
I know there are the trivial subspaces.
I thought the subspaces may be the possible lines ( I mean the possible inclinations )
For example, would have the lines whose parallel vectors are (0,1) (0,2) ... (0,6) (1,0) (1,1) (1,2) (1,3) ... (1,6) (2,0) (2,1) (2,3) (2,5) etc.
Is it right?
That is correct. If a subspace S is not the zero subspace then it must contain a nonzero element (x,y). If x=0 then . If x≠0 then , so S contains all the elements . If S contains other elements than those, then you should be able to prove by similar arguments that S is the whole of .
Edit: That Escher avatar is TOO BIG.