This document provides an overview of linear systems of equations including: 1. Definitions of linear equations and systems of linear equations. A linear system is a set of linear equations with the same variables. 2. Methods for solving linear systems including row operations like Gauss elimination and inverse matrix methods. 3. Concepts like the number of solutions a linear system can have, whether a system is consistent or inconsistent, and the use of augmented and coefficient matrices to represent systems.

1. Lecture 2
LINEAR SYSTEM OF EQUATIONS
Learning outcomes: by the end of this lecture
1. You should know,
a)What is a linear system of equations
b)What is a homogeneous system
c)How to represent a linear system in matrix form
d) What is a coefficient matrix
e)What is an augmented matrix
2. You should be able to solve a linear system of
equationsusing:
a)Row operations:
i. Gauss-elimination method (REF)
ii. Gauss-Jordan method (RREF)
b) Inverse matrix method

2. Definitionof a LinearEquationin n Variables:
A linearequationin n variable nxxx ,,, 21  has the form
bxaxaxa nn  2211
Where the coefficients baaa n ,,,, 21  arereal numbers
(usuallyknown). The number of 1a is the leading
coefficientand 1x is the leadingvariable.
The collectionof several linear equationsis referred to as
the systemof linear equations.
Definitionof System of m LinearEquationin n
Variables:
A system of m linearequationsin n variablesis a set
of m equations,each of which is linearin the same n
variables:
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa







2211
22222121
11212111
where ,,1,,,2,1,, njmiba iij   are constants.

3. Example:1Consider the following system of linear
equations:
67
832
3,42523
43
31
321
321
321




xx
xxx
nmxxx
xxx
Example: 2 Which of the following are linear equations?
2
1 2 3
1 2 3
( ) 3 2 7 ( ) (sin ) 4 (log5) ( ) 2
3
1
( ) 2 4 ( ) sin 2 3 0 ( ) 4x
a x y b x x x e c x y
d e y e x x x f x
y

      
      
( ) and ( ) are linear equations.a b ( ),( ),( ), and ( ) are not linear.c d e f
 Number of Solutionsof a System of Linear
Equations
Consider the following systems of linear equations
(a)
1
3
x y
x y
 

 
(b)
4
2
x y
x y
 

 
(c)
6 2 8
3 4
x y
x y
   

 
For a system of linear equations, precisely one of the
following is true:
(a) The system has exactly one solution.
(b) The system has no solution.
(c) The system has infinitely many solutions.
1x y 
3x y  2x y 
4x y 

4.  Consistent and Inconsistent
A system of linear equations is called consistent if it has
at least one solution and inconsistent if it has no solution.
 Equivalent
Two systems of linear equations are said to be equivalent
if they have the same set of solutions.
 Back – Substitution
Which of the following systems is easier to solve?
2 3 9
( ) 3 7 6 22
2 5 5 17
x y z
a x y z
x y z
  

    
   
2 3 9
( ) 3 5
2
x y z
b y z
z
  

 
 
System (b) is said to be in row-echelon form. To solve
such a system, use a procedurecalled back – substitution.
 Augmented Matrices and Coefficient Matrices
Consider the m n linear system
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2 ...
n n
n n
m m mn n m
a x a x a x b
a x a x a x b
a x a x a x b
   
   
   
Let
 
11 12 1
21 22
1 1
2
11 12 1
21 22 2
2 1 2
22
1
, , |
n
n
n
n
m m m m mm m mn n
b b
b
a a a
a a a
A
a a a
a a a
a a
b B Ab
a ba
b
aba
 
 
 
 
 
 
 
 
 
 
 

 
 
    
 
 
 
A is called the coefficient matrix of the system.

5. B is called the augmented matrix of the system.
b is called the constant matrix of the system.
It is possibleto write the system
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2 ...
n n
n n
m m mn n m
a x a x a x b
a x a x a x b
a x a x a x b
   
   
   
in the followingmatrix form
BXA
b
b
b
m
2
1
2
1
21
22221
11211










































nmnmm
n
n
x
x
x
aaa
aaa
aaa
Example: 65y-2
13


x
yx
BXA
6
1
y
x
52
13





















 Row-Equivalent
Two m n matrices are said to be row-equivalent if one can
be obtained by the other by a series of elementary row
operations.