Lecture 6 - Linear Transformations | Linear Algebra - Engineering Mathematics PDF Download

 Page 1


                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
 
                                                                                         
                                                    
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Linear Transformations 
Lesson Developer: Dr. Arvind and Itendra kumar 
College: Hansraj College, University of Delhi 
 
                                       
 
 
 
 
 
 
 
 
Page 2


                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
 
                                                                                         
                                                    
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Linear Transformations 
Lesson Developer: Dr. Arvind and Itendra kumar 
College: Hansraj College, University of Delhi 
 
                                       
 
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
Table of contents: 
Chapter: Linear Transformations 
      1- Learning outcomes 
      2- Introduction 
      3- Linear transformations 
      4- Some special types of linear transformations 
      5- Kernel and Range of a Linear Transformation 
      6- Basis and Dimension Theorem 
      7- Singular and Nonsingular Linear Transformations 
      8- Isomorphism 
      9- Operations with Linear Transformations 
      10- Algebra of Linear Transformations 
      11- Exercise 
      12- References 
 
 
 
 
 
 
 
 
Page 3


                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
 
                                                                                         
                                                    
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Linear Transformations 
Lesson Developer: Dr. Arvind and Itendra kumar 
College: Hansraj College, University of Delhi 
 
                                       
 
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
Table of contents: 
Chapter: Linear Transformations 
      1- Learning outcomes 
      2- Introduction 
      3- Linear transformations 
      4- Some special types of linear transformations 
      5- Kernel and Range of a Linear Transformation 
      6- Basis and Dimension Theorem 
      7- Singular and Nonsingular Linear Transformations 
      8- Isomorphism 
      9- Operations with Linear Transformations 
      10- Algebra of Linear Transformations 
      11- Exercise 
      12- References 
 
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
1. Learning Outcomes 
After study of this chapter you will be  able to understand: 
 I- What is Linear Mappings and its applications. 
 II-  How to find out Kernel and Range of a Linear Transformation. 
 III-The study of cases to find out the linear transformation is singular  
      or non singular. 
 IV-  When we say Linear transformation is an isomorphism.. 
 V- Geometrical applications of  Linear mappings. 
 VI-  Composition of Linear mappings. 
 VII-  Basis and Dimension of a linear Transformations. 
 VIII- Some important operations with linear transformations. 
 IX- Some good examples related to all topics. 
 X-  Algebra of Linear mappings. 
 XI- In References we mentioned some good books of linear algebra. 
          
 
 
 
 
 
 
 
Page 4


                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
 
                                                                                         
                                                    
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Linear Transformations 
Lesson Developer: Dr. Arvind and Itendra kumar 
College: Hansraj College, University of Delhi 
 
                                       
 
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
Table of contents: 
Chapter: Linear Transformations 
      1- Learning outcomes 
      2- Introduction 
      3- Linear transformations 
      4- Some special types of linear transformations 
      5- Kernel and Range of a Linear Transformation 
      6- Basis and Dimension Theorem 
      7- Singular and Nonsingular Linear Transformations 
      8- Isomorphism 
      9- Operations with Linear Transformations 
      10- Algebra of Linear Transformations 
      11- Exercise 
      12- References 
 
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
1. Learning Outcomes 
After study of this chapter you will be  able to understand: 
 I- What is Linear Mappings and its applications. 
 II-  How to find out Kernel and Range of a Linear Transformation. 
 III-The study of cases to find out the linear transformation is singular  
      or non singular. 
 IV-  When we say Linear transformation is an isomorphism.. 
 V- Geometrical applications of  Linear mappings. 
 VI-  Composition of Linear mappings. 
 VII-  Basis and Dimension of a linear Transformations. 
 VIII- Some important operations with linear transformations. 
 IX- Some good examples related to all topics. 
 X-  Algebra of Linear mappings. 
 XI- In References we mentioned some good books of linear algebra. 
          
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
 
2. Introduction 
      The goal of this chapter is study of linear mappings or linear 
transformations. Linear mapping is a function whose domain and range, sets 
are subsets of vector spaces or linear mapping is function from a vector 
space into vector space. The linear transformation denoted by  
                                         
T : U W ?
 
     The above symbol denote that T is a function whose domain is U and 
whose range set is W. For each element a in U, the element T(a) in W is 
called the image of a under T, and generally we say that T maps a into T(a).  
     If B is any subset of U, the set of all images T(a) for a in B is called the 
image of  B under T and is  denoted by T(B). The image of the domain U, 
T(U),  is the range T 
3. Linear Transformations 
Definition:  Let U and V be vector spaces over the field F. A linear mapping 
from U into V is a function T from U into V such that   
 T(c d ) cT( ) dT( ) ? ? ? ? ? ? ? 
for all and ?? in U and all scalar c and d in F. 
Example 1: Let K be a field and let U be the space of polynomial functions g 
from K into K, given by 
 
k
0 1 k
g(x) a a x .... a x ? ? ? ? 
Let 
 
k1
1 2 k
(Dg)(x) a 2a x ... ka x
?
? ? ? ? 
Page 5


                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
 
                                                                                         
                                                    
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Linear Transformations 
Lesson Developer: Dr. Arvind and Itendra kumar 
College: Hansraj College, University of Delhi 
 
                                       
 
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
Table of contents: 
Chapter: Linear Transformations 
      1- Learning outcomes 
      2- Introduction 
      3- Linear transformations 
      4- Some special types of linear transformations 
      5- Kernel and Range of a Linear Transformation 
      6- Basis and Dimension Theorem 
      7- Singular and Nonsingular Linear Transformations 
      8- Isomorphism 
      9- Operations with Linear Transformations 
      10- Algebra of Linear Transformations 
      11- Exercise 
      12- References 
 
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
1. Learning Outcomes 
After study of this chapter you will be  able to understand: 
 I- What is Linear Mappings and its applications. 
 II-  How to find out Kernel and Range of a Linear Transformation. 
 III-The study of cases to find out the linear transformation is singular  
      or non singular. 
 IV-  When we say Linear transformation is an isomorphism.. 
 V- Geometrical applications of  Linear mappings. 
 VI-  Composition of Linear mappings. 
 VII-  Basis and Dimension of a linear Transformations. 
 VIII- Some important operations with linear transformations. 
 IX- Some good examples related to all topics. 
 X-  Algebra of Linear mappings. 
 XI- In References we mentioned some good books of linear algebra. 
          
 
 
 
 
 
 
 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
 
 
2. Introduction 
      The goal of this chapter is study of linear mappings or linear 
transformations. Linear mapping is a function whose domain and range, sets 
are subsets of vector spaces or linear mapping is function from a vector 
space into vector space. The linear transformation denoted by  
                                         
T : U W ?
 
     The above symbol denote that T is a function whose domain is U and 
whose range set is W. For each element a in U, the element T(a) in W is 
called the image of a under T, and generally we say that T maps a into T(a).  
     If B is any subset of U, the set of all images T(a) for a in B is called the 
image of  B under T and is  denoted by T(B). The image of the domain U, 
T(U),  is the range T 
3. Linear Transformations 
Definition:  Let U and V be vector spaces over the field F. A linear mapping 
from U into V is a function T from U into V such that   
 T(c d ) cT( ) dT( ) ? ? ? ? ? ? ? 
for all and ?? in U and all scalar c and d in F. 
Example 1: Let K be a field and let U be the space of polynomial functions g 
from K into K, given by 
 
k
0 1 k
g(x) a a x .... a x ? ? ? ? 
Let 
 
k1
1 2 k
(Dg)(x) a 2a x ... ka x
?
? ? ? ? 
                   Linear Transformations  
 
Institute of Lifelong Learning, University of Delhi                                            
 
Then D is a linear transformation from U into U the differentiation 
transformation. 
Theorem 1: Let V be a finite-dimensional vector space over the field K and 
let 
1n
{ ..., } ?? be an ordered basis for V. Let U be a vector space over the 
same field K and let 
1n
,..., ?? be any vectors in U. Then there is precisely one 
linear transformation T from V into U such that 
 
ii
T , i 1,...,n. ? ? ? ? 
Proof:  To prove there is some linear transformation T with 
ii
T ? ? ? . For 
given ? in V, there is a unique n-tuple 
1n
(x ,...,x ) such that 
 ? 
1 1 n n
x .... x ? ? ? ? ? 
For this vector ? we define  
 
1 1 n n
T x ... x ? ? ? ? ? ? 
Then T is a well-defined rule for associating with each vector ? in V a vector 
T ? in U. From the definition it is clear that 
ii
T ? ? ? for each i. To see that T is 
linear, let 
 ?
1 1 n n
y .... y ? ? ? ? ? 
be in V and let c be any scalar. Now 
 c ? ? ?
1 1 1 n n n
(cx y ) ... (cx y ) ? ? ? ? ? ? ? 
and so by definition 
 T(c ) ? ? ?
1 1 1 n n n
(cx y ) ... (cx y ) ? ? ? ? ? ? ? 
On the other hand, 
 c(T ) T ? ? ?
nn
i i i i
i 1 i 1
c x y
??
? ? ? ?
??