Let F_2 be the field with two elements and F_256 the field with 256 elements.
1) Proof that there is no subset M of (F_2)^8 with card(M)=10 so that all subsets N of M with card(N)=8 are linearly independent.
2) Change in 1) F_2 with F_256 and proof that unlike in (F_2)^8 there are such subsets M.

So I think I solved 1): Let v_i 0<i<9 be linearly independet vectors in V:=(F_2)^8. To form a subset M with card(M)=10 we need another two vectors that are now called w1 and w2. These are also linearly indepent (otherwise w1=w2 or one w=0). Write these ten vectors in a matrix (first v then w) and reduce it. (The linear indepence remains unchanged.) Then you have the 8x8 identity matrix and two more columnes. These two have at least one 0 because they are linearly independent. So take the set of v and replace one v with a w so that the w has in the reduced matrix a 0 where the 1 of the v is. This subset should be linearly dependent.

I hope that's right. But I have no idea how to do 2). Clearly it has a bit more elements...
Maybe it's possible to find an M that redeems the terms. However it should also be possible to formerly proof that there must be such a subset M, but how?