An Underdetermined Linear System for GPS
Dan Kalman (email@example.com) joined the
mathematics faculty at American University in 1993,
following an eight year stint in the aerospace industry and
earlier teaching positions in Wisconsin and South Dakota.
He has won three MAA writing awards and served a term
as Associate Executive Director of the MAA. His interests
include matrix algebra, curriculum development, and
interactive computer environments for exploring
mathematics, especially using Mathwright software.
Finding the general solution to an underdetermined linear system is a standard topic
in linear algebra. It contributes to a complete analysis of the behaviors of linear sys-
tems, as well as providing a foundation for understanding more abstract topics, includ-
ing linear transformations, null space, and dimension. But at the first introduction of
the topic it would be nice to have a simple, realistic example where the parameterized
general solution of an underdetermined system is of practical interest. In this note, I
will present such an example connected with the Global Positioning System (GPS) for
determining geographical locations.
The basic idea of GPS is a variant on three dimensional triangulation: a point on
the surface of the earth is determined by its distances from three other points. Here,
the point we wish to determine is the location of the GPS receiver, the other points are
satellites, and the distances are computed using the travel times of radio signals. This
requires accurate time keeping, prompting a slight modification of the pure spatial tri-
angulation problem. In the modified version, we need four satellites, rather than three,
and can then calculate both the location, and the correct time, at the GPS receiver.
Before presenting the example, I should make it clear that the computations that fol-
low are not the same as the methods actually used by GPS. The example assumes exact
geometric knowledge, whereas GPS has to deal with real world measurement errors.
Thus, GPS typ