Real-world Application of Linear Algebra

The Earth travels around the Sun in an ellipse, and the Sun is at a focal point of that ellipse. Likewise for a satellite travelling around the Earth (Figrue 1). It is possible for a satellite to travel in a circular orbit, but that is a special case. From such information we are able to derive the orbit of an object using linear algebra. The satellite-based Global Positioning System
(GPS) works by locking on to the signals of at least three satellites to calculate a 2 dimensional
position (latitude and longitude) and track movement. With four or more satellites in view, the receiver can determine the userʼs 3D position (latitude, longitude and altitude). The location
of an object could be identifi ed using 3 satellites or more. This can be done by solving linear
equations.
Mixing chemicals is a method used by a chemist to provide a useful outcome (e.g. solid, gas, liquid, spark, perfume... etc.) or not useful outcome (e.g. fire, bad smell, explosion... etc.). Dealing with chemicals is dangerous, but mixing chemicals is even more dangerous. The chemist has to
ensure that s/he is using the right amount of each chemical or there may be a disaster (or by luck a new invention). For example, when developing photographic negatives, s/he is dealing with three chemicals (developer, fixer and stop path), each one to be mixed with water in amounts specified by the manufacturer. Incorrect mixing will spoil your nice negatives. Where there is a large number of chemicals involved, computer packages can help to determine the right amounts to be included in the mix. Those computer packages use a technique that involves linear algebra.

Figure 1. Satellite communications and the satellite-
based Global Positioning System (GPS) have
benefi tted from mathematical models using linear
algebra.


Figure 2. Chemical mixing can be modelled using
linear algebra.
http://physics.dit.ie/resources/yearbook2005/alkalbani.pdf