If u have a 4x3 matrix that is linearly independent, why can't you solve for Ax = b for EVERY b ?
The point made before is that a n by m matrix, with independent columns, maps to an n dimensional subspace of . If n< m, that is NOT all of and there will be vectors b not in that subspace so Ax= b would have no solution.
But if n= m, that is, if A is an n by n matrix with independent columns, then its image is all of .