Difference between least squares and minimum norm solution Consider a linear system of equations $Ax = b$.


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*If the system is overdetermined, the least squares (approximate) solution minimizes $||b - Ax||^2$. Some source sources also mention $||b - Ax||$.

*If the system is underdetermined one can calculate the minimum norm solution. But it does also minimize  $||b - Ax||$, or am I wrong?
But if least squares is also a minimum norm, what is the difference, or the rationale of the different naming?

If a system is overdetermined, there is no solution and thus we may want to find $x$ such that $||Ax-b||$ is as small as it can be (as there is no way to make $||Ax-b||=0$).
On the other hand, if the system is underdetermined, there are infinitely many solutions and thus one can find a solution of minimal norm and this is called the minimum norm solution. 
