Let us look at some examples of linear systems.

- Suppose
Consider the system
- If then the system has a UNIQUE SOLUTION
- If
and
- then the system has NO SOLUTION.
- then the system has INFINITE NUMBER OF SOLUTIONS, namely all

- 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- UNIQUE SOLUTION

and The unique solution is

Observe that in this case, - 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 - 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

- UNIQUE SOLUTION
- 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.- UNIQUE SOLUTION

Consider the system and The unique solution to this system is*i.e.*THE THREE PLANES INTERSECT AT A POINT. - 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. - 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.The readers are advised to supply the proof.

- UNIQUE SOLUTION

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).

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

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.