In linear algebra (linear inverse problems) one generalizes the notion of a solution of a linear operator equation $Ax=y$ to

1. "best approximation" if there is no solution, i.e. minimizing the functional $\|Ax-y\|$,
2. "Minimum-norm solution" if there is a subspace of solutions, i.e. taking that solution of $Ax=y$ which has minimal norm,
3. both (if the best approximation is not unique) leading to the Moore-Penrose inverse.