Linear algebra applications in GPS and orbital mechanics

Algebra

Like many other languages and sciences, Algebra can be useful in the explanation of real-world experiences. Linear algebra, in particular, holds a high level of relevancy in the solution of real world problems like physics equations. Since the key point of physics is to explain the world in proven observations, linear algebra is an ideal mode for discussion. Many real-world situations can be explained by algebra; for example, how does GPS work? The satellite-based Global Positioning System works by locking onto the system of three satellites and calculating a two-dimensional position from latitude and longitude, thus tracking movement. The location of objects can be determined by using linear equations to morph the data into identifiable locations, and with four or more satellites in view, altitude combines with latitude and longitude to determine the 3-D position.

A far more generic (but equally important) use of linear algebra in real world discussions is used to explain the law of motions as described by Keplar. Keplar's laws relate to the orbit of an object moving around another in space as elliptical, with the stationary object located at one of the focal points of the ellipse. Simply put, the Earth travels around the sun in an ellipse, and the sun is a focal point in that ellipse; likewise for a satellite traveling around the earth.

Using Algebra, Keplar understood: