Lecture 6 - Systems of Linear Equation | Algebra- Engineering Maths - Engineering Mathematics PDF Download

 Page 1


System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Systems of Linear Equations 
Course Developer: Gurudatt Rao Ambedkar 
College: Acharya Narendra Dev College, (D.U.) 
 
 
 
 
 
 
 
Page 2


System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Systems of Linear Equations 
Course Developer: Gurudatt Rao Ambedkar 
College: Acharya Narendra Dev College, (D.U.) 
 
 
 
 
 
 
 
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 2 
 
 
  
Page 3


System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Systems of Linear Equations 
Course Developer: Gurudatt Rao Ambedkar 
College: Acharya Narendra Dev College, (D.U.) 
 
 
 
 
 
 
 
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 2 
 
 
  
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 3 
 
 
 
Table of Contents 
 Chapter : System of linear equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: System of linear equation 
? 3.1: Linear equation 
? 3.2: Solution of linear equation 
? 3.3: System of linear equation 
? 3.4: Homogeneous/non-homogeneous system of linear 
equation 
? 3.5: Type of system of linear equation and its solution 
? 3.6: Particular solution 
? 4: Echelon form 
? 4.1: Pivot and free variable 
? 4.2: Solution of an echelon form 
? 5: Augmented matrix of a linear system 
? 6: Reduced row Echelon form of matrix 
? 7: Row reduction theorem 
? 8: Vector equation 
? 9: Linear combination of vectors 
? 10: Linear Span 
? 11:  Matrix form of linear system 
? 11.1:  Matrix equation 
? 11.2:  Properties of matrix-vector Product AX 
? 12:  Solution of a system of linear equation in vector form 
? 12.1:  Homogeneous system of linear equation 
Page 4


System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Systems of Linear Equations 
Course Developer: Gurudatt Rao Ambedkar 
College: Acharya Narendra Dev College, (D.U.) 
 
 
 
 
 
 
 
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 2 
 
 
  
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 3 
 
 
 
Table of Contents 
 Chapter : System of linear equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: System of linear equation 
? 3.1: Linear equation 
? 3.2: Solution of linear equation 
? 3.3: System of linear equation 
? 3.4: Homogeneous/non-homogeneous system of linear 
equation 
? 3.5: Type of system of linear equation and its solution 
? 3.6: Particular solution 
? 4: Echelon form 
? 4.1: Pivot and free variable 
? 4.2: Solution of an echelon form 
? 5: Augmented matrix of a linear system 
? 6: Reduced row Echelon form of matrix 
? 7: Row reduction theorem 
? 8: Vector equation 
? 9: Linear combination of vectors 
? 10: Linear Span 
? 11:  Matrix form of linear system 
? 11.1:  Matrix equation 
? 11.2:  Properties of matrix-vector Product AX 
? 12:  Solution of a system of linear equation in vector form 
? 12.1:  Homogeneous system of linear equation 
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 4 
 
? 13:  Linear Independence 
? 14:  Linear dependence 
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
1. Learning Outcomes 
After studying this chapter, you should be able to 
? classify homogeneous and non homogeneous system of linear 
equations; 
? change system of linear equations in  matrix equation/vector equation; 
? change matrices into echelon and reduced row echelon form; 
? obtain the general solution of system of linear equations; 
? obtain and classify different type of solutions of linear system; 
? understand the concept of linear independency of vectors;                                                                                                        
 
 
 
 
 
 
 
 
Page 5


System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Systems of Linear Equations 
Course Developer: Gurudatt Rao Ambedkar 
College: Acharya Narendra Dev College, (D.U.) 
 
 
 
 
 
 
 
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 2 
 
 
  
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 3 
 
 
 
Table of Contents 
 Chapter : System of linear equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: System of linear equation 
? 3.1: Linear equation 
? 3.2: Solution of linear equation 
? 3.3: System of linear equation 
? 3.4: Homogeneous/non-homogeneous system of linear 
equation 
? 3.5: Type of system of linear equation and its solution 
? 3.6: Particular solution 
? 4: Echelon form 
? 4.1: Pivot and free variable 
? 4.2: Solution of an echelon form 
? 5: Augmented matrix of a linear system 
? 6: Reduced row Echelon form of matrix 
? 7: Row reduction theorem 
? 8: Vector equation 
? 9: Linear combination of vectors 
? 10: Linear Span 
? 11:  Matrix form of linear system 
? 11.1:  Matrix equation 
? 11.2:  Properties of matrix-vector Product AX 
? 12:  Solution of a system of linear equation in vector form 
? 12.1:  Homogeneous system of linear equation 
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 4 
 
? 13:  Linear Independence 
? 14:  Linear dependence 
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
1. Learning Outcomes 
After studying this chapter, you should be able to 
? classify homogeneous and non homogeneous system of linear 
equations; 
? change system of linear equations in  matrix equation/vector equation; 
? change matrices into echelon and reduced row echelon form; 
? obtain the general solution of system of linear equations; 
? obtain and classify different type of solutions of linear system; 
? understand the concept of linear independency of vectors;                                                                                                        
 
 
 
 
 
 
 
 
System of Linear Equations 
        Institute of Lifelong Learning, University of Delhi                                                      
pg. 5 
 
 
 
2. Introduction: 
System of linear equation play important role in many problems of the real 
world. System of linear equations is very useful to solve many problem of 
mathematics specially of linear algebra. In this chapter we are introducing a 
brief idea about the concept and properties of linear equation and their 
system.  
Coefficients and constants used in all our system of linear equations comes 
from any number field K.  In general, scalars used in this chapter are real 
numbers, that is, they come from the real field R. 
Figure 1 in Section 3.5 present a systematic diagram of systems of linear 
equations and their solutions. Section 8 and 11 shows that how a system of 
linear equations can be equivalent to a vector equation and to a matrix 
equation. This equivalence will useful to reduce the systems of linear 
equation into linear combinations of vectors. The basic concepts of linear 
independence, spanning, also introduced in this chapter.   
3. System of linear equations: 
3.1. Linear equation: 
An equation in n unknown 
1
x
,
2
x
,
3
x
, . . . , 
n
x is said to be a linear equation if it 
can be represented in the given standard form
: 
b x c x c x c x c
n n
? ? ? ? ? . . .
3 3 2 2 1 1
 
Where
1
c
 
,
2
c , . . ., 
n
c and b are constant. The constant c
i 
is called the 
coefficient of
i
x
.