# Introduction

Let us look at some examples of linear systems.

1. Suppose Consider the system
1. If then the system has a UNIQUE SOLUTION
2. If and
1. then the system has NO SOLUTION.
2. then the system has INFINITE NUMBER OF SOLUTIONS, namely all
2. We now consider a system with equations in unknowns.
Consider the equation If one of the coefficients, or is non-zero, then this linear equation represents a line in Thus for the system

the set of solutions is given by the points of intersection of the two lines. There are three cases to be considered. Each case is illustrated by an example.
1. UNIQUE SOLUTION
and The unique solution is
Observe that in this case,
2. INFINITE NUMBER OF SOLUTIONS
and The set of solutions is with arbitrary. In other words, both the equations represent the same line.
Observe that in this case, and
3. NO SOLUTION
and The equations represent a pair of parallel lines and hence there is no point of intersection.
Observe that in this case, but
3. As a last example, consider equations in unknowns.
A linear equation represent a plane in provided As in the case of equations in unknowns, we have to look at the points of intersection of the given three planes. Here again, we have three cases. The three cases are illustrated by examples.
1. UNIQUE SOLUTION
Consider the system and The unique solution to this system is i.e. THE THREE PLANES INTERSECT AT A POINT.
2. INFINITE NUMBER OF SOLUTIONS
Consider the system and The set of solutions to this system is with arbitrary: THE THREE PLANES INTERSECT ON A LINE.
3. NO SOLUTION
The system and has no solution. In this case, we get three parallel lines as intersections of the above planes taken two at a time.

DEFINITION 2.1.1 (Linear System)   A linear system of equations in unknowns is a set of equations of the form

 (2.1.1)

where for and Linear System (2.1.1) is called HOMOGENEOUS if and NON-HOMOGENEOUS otherwise.

We rewrite the above equations in the form where
and

The matrix is called the COEFFICIENT matrix and the block matrix is the AUGMENTED matrix of the linear system (2.1.1).

Remark 2.1.2   Observe that the row of the augmented matrix represents the equation and the column of the coefficient matrix corresponds to coefficients of the variable That is, for and the entry of the coefficient matrix corresponds to the equation and variable

For a system of linear equations the system is called the ASSOCIATED HOMOGENEOUS SYSTEM.

DEFINITION 2.1.3 (Solution of a Linear System)   A solution of the linear system is a column vector with entries such that the linear system (2.1.1) is satisfied by substituting in place of

That is, if then holds.

Note: The zero -tuple is always a solution of the system and is called the TRIVIAL solution. A non-zero -tuple if it satisfies is called a NON-TRIVIAL solution.

Subsections
A K Lal 2007-09-12