Yes, you are right- those three vectors are independent and so form a basis for their span.
I'm trying to determine if S is a basis for V if S={(1,2,1,0),(2,1,3,1),(2,2,1,2)} in V=spanS
so I made the matrix and put it in (not reduced) row echelon form which gives and this means that (1,2,1,0),(2,1,3,1) and (2,2,1,2) form a basis for and the dimension is 3.
I have a feeling I'm doing something wrong. Can someone tell me if this is right?