Do you know about "PseudoInverse" or "Moore-Penrose Inverse"? This is based on "SingularValueDecomposition". If you do not know this terms, look them up.
For your example:
res = PseudoInverse[m] . b; Now to check how accurate this is;
Max[Abs[m . res - b]] 1.97774269073*10^-7 Is this accurate enough for your application?