Okay but then why does a 4X4 matrix that is linearly independent with Ax= b can be solved for every b? i don't see how you use the fact that the matrix in question was 4x3
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 .