A *m × n* system of linear equations where
*m > n* (or *m = n* and **A** is singular)
is underdetermined. There are fewer equations than there
are unknowns. Underdetermined systems have *q* linearly
independent families of solutions, where

$$q=n-r$$ |

and

$$r=\operatorname{rank}\left(\mathbf{A}\right)$$ |

The value *q* is referred to as the nullity of matrix
**A**. The *q* linearly dependent equations in
**A** are the null space of **A**.

”Solving” an underdetermined set of equations usually boils
down to solving a fully determined *r × r* system
(known as the range of **A**) and adding this solution
to any linear combination of the other *q* vectors of
**A**. A numerical procedure that solves the crux of
this problem is known as singular value decomposition (or
SVD). A singular value decomposition constructs a set of
orthonormal bases for the null space and range of **A**.