linear system

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.