Let A denote an mxn matrix and let A' denote the row echelon form of it, which has d steps. We then have according to my textbook:
1) If m>d there exists a column such that the set of equation has no solutions.
2) If n>d the matrixequation AX=0 has a set of solution expressed parametrically by (n-d) parameters.
3) If m=n=d there exists a unique solution for every B in the equation AX = B.
Now 1) and 3) I understand. What troubles me is 2). Why to they switch the equation to AX=0 rather than AX=B? Wouldn't that last equation also be dependent on n-d parameters. I'm pretty sure that this has something to do with the fact that the nullspace is quite a unique thing since it forms a linear subspace. Thus we can define its dimension and later use all this to prove the rank nullity theorem. So can someone explain what's going on on a deeper level?