Chapter 4 System of Linear Equations Section 4.1 What are Systems of Linear Equations?
Before we can really start, let us clarify the terminology.
Several equations relating a specific number of variables at the same time form a so-called system of equations. If the variables in every equation occur only linearly, i.e. at most to the power of , and are only multiplied by (constant) numbers, the system is called a system of linear equations, or LS (linear system).
The two equations in the first example 4.1.1 form a system of linear equations in the variables and . In contrast, the three equations
do form a system of equations in the variables , , and , but the system is not linear since in the third equation the term occurs, which is bilinear in the variables and and hence violates the condition of linearity.
By the way, in a system of equations the number of equations need not be equal to the number of variables; we will return to this later on.
Systems of linear equations are distinguished from general systems of equations by their relative simplicity. Nevertheless, they play an important role in fields as diverse as medical science (e.g. for CT scans), engineering (e.g. for describing sound propagation in complex designed spaces), or physics (e.g. concerning the question of which wave lengths excited atoms can emit). It is, without doubt, worth dealing with systems of linear equations intensely.
Which of the following systems is a system of linear equations?
|, , and ,|
For systems of equations generally, the question focuses on which values the variables must take such that all equations of the system are simultaneously satisfied. Such a set of values for the variables is called a solution of a system of equations.
Before we solve systems of equations a detail should be noted: depending on the problem, it may not be useful to accept all variable values. In the first example 4.1.1 the variables and are the numbers of unicycles and bicycles the group of stuntmen owns. Such numbers can only be non-negative integers, i.e. elements of . Hence, in this case the number range for the solutions has to be restricted to in advance (namely, for both and ).
The possible number range for the solutions of a system of equations is called the base set of the system. The domain is the subset of the base set for which all the terms of the equations of the system are defined. For systems of linear equations, base set and domain coincide. Finally, the solution set is the subset of the domain which merges the solutions of the system. The solution set is denoted by .