Linear Equations – Preview

   A linear equation in   unknowns is an expression which can be written as

   

We explain the meaning to the symbols in this equation:

  • , ... ,   represent numbers which are given. They are called the coefficients of the equation.
  •   on the right represents a number which is also given. The equation is called homogeneous if  0, and is called inhomogeneous if  0.        
  • , ... ,   are variables or "unknowns", and our task is to determine number values for  , ... ,   such that the given equation is true.

More generally, we will need to determine simultaneous solutions of a system of linear equations in    unknowns. Fortunately, there are several methods at our disposal to respond to this task. In this chapter we discuss the following:

   A geometrical method:  It uses the fact that the solutions of a single linear equation in    unknowns form a hyperplane in  . Therefore the simultaneous solutions of such equations form the intersection of the corresponding hyperplanes. - This method enables us to discuss qualitatively the solutions of a system of linear equations.

   The elimination method of Gauß and Jordan yields explicitly the solutions of a given system of linear equations. This method is a programmable procedure which transforms a given system of linear equations into simpler and simpler ones. In the end, one is left with a system of linear equations which is so simple that its solutions can be read off immediately.

   Later we will learn another method for solving certain kinds of systems of linear equations: Cramer’s rule is a formula which is applicable to non-degenerate systems of equations with unknowns. Such a system has always exactly one solution, and Cramer’s rule computes it by a formula involving determinants.