question about procedure in determining if vectors form a basis

If I'm given a set of vectors and asked to determine whether they form a basis of a certain subspace, I check to see if they are linearally independent and check to see if they span the subspace. So isn't that doing the same work twice? What I mean is that to check to see if they are linearally independent I'd put them into rref and if there's anything asside from 1s and 0s they are linearally dependent. For them to span the subspace they I would put the vectors in rref and basically if they don't form the identitity matrix they don't span the subspace. So putting it simply, isn't it enough to take the matrix, put it in rref, and if it doesn't give the identity matrix (unless there's extra rows of 0s on the bottom) then it's doesn't form a basis?