Open App

Engineering Mathematics Exam > Engineering Mathematics Notes > Algebra- Engineering Maths > Lecture 7 - Linear Transformations

Lecture 7 - Linear Transformations | Algebra- Engineering Maths - Engineering Mathematics PDF Download

Download, print and study this document offline

Please wait while the PDF view is loading

 Page 1


Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Algebra-I 
Lesson: Linear Transformations  
Lesson Developer: Chaman Singh  
College/Department: Acharya Narendra Dev College (D.U.)  
 
 
 
 
 
 
 
 
 
 
 
Page 2


Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Algebra-I 
Lesson: Linear Transformations  
Lesson Developer: Chaman Singh  
College/Department: Acharya Narendra Dev College (D.U.)  
 
 
 
 
 
 
 
 
 
 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 2 
 
 
 
Table of Contents: 
 Chapter : Linear Transformations 
? 1. Learning Outcomes 
? 2. Linear Transformation 
? 3. Matrix Transformation 
o 3.1. Contraction and Dilation 
o 3.2. Shear Transformation 
? 4. Matrix of a Linear Transformation 
o 4.1. Basis 
o 4.2. Steps to find the matrix of the transformation  
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
1. Learning Outcomes: 
After studying the whole contents of this chapter, students will be able to 
understand:  
? Linear transformation 
? Matrix transformation 
? Contraction and Dilation 
? Shear Transformation 
? Matrix of linear transformations 
? Basis of linear transformations 
? How to find the matrix of a linear transformation 
 
 
 
Page 3


Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Algebra-I 
Lesson: Linear Transformations  
Lesson Developer: Chaman Singh  
College/Department: Acharya Narendra Dev College (D.U.)  
 
 
 
 
 
 
 
 
 
 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 2 
 
 
 
Table of Contents: 
 Chapter : Linear Transformations 
? 1. Learning Outcomes 
? 2. Linear Transformation 
? 3. Matrix Transformation 
o 3.1. Contraction and Dilation 
o 3.2. Shear Transformation 
? 4. Matrix of a Linear Transformation 
o 4.1. Basis 
o 4.2. Steps to find the matrix of the transformation  
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
1. Learning Outcomes: 
After studying the whole contents of this chapter, students will be able to 
understand:  
? Linear transformation 
? Matrix transformation 
? Contraction and Dilation 
? Shear Transformation 
? Matrix of linear transformations 
? Basis of linear transformations 
? How to find the matrix of a linear transformation 
 
 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 3 
 
 
 
2. Linear Transformation: 
Consider the Euclidean spaces 
nm
R and R . A mapping :
nm
T R R ?  is called 
the linear mapping or linear transformation if T satisfies the following two 
axioms: 
 
( ) ( ) ( ) ( )
( ) ( ) ( )
I T x y T x T y
II T ax aT x
? ? ?
?
   
Where all ,
n
x y R ?  and all aR ? . 
Value Addition: Note 
The set 
n
R is said to be the domain of T, and 
m
R co-domain of T. 
 
Value Additions: Do you know? 
A mapping or function from X to Y i.e. : f X Y ?  is a rule that assigns to 
each element x in X a unique element y in Y. The set X is called the 
domain of f and the set Y is called the co-domain of f. For , xX ? the 
element yY ? , such that f(x) = y, is called the image of x (under the 
action of f) and the set of all images yY ? is called the range of f. 
 
Value Addition: Remember 
Note: a mapping :
nm
T R R ? is linear if it "preserves" the two basic 
operations of a vector space, 
(i) vector addition, i.e., ( ) ( ) ( ), , ( ), ( )
nm
T x y T x T y x y R andT x T y R ? ? ? ? ? ? , 
addition of x and y on left is the addition of vectors of 
n
R and addition of 
T(x) and T(y) on right is addition of vectors of 
m
R . 
(ii) scalar multiplication, i.e.,  
( ) ( ),
n
T ax aT x x R and a R ? ? ? ? ?  
Scalar multiplication on left i.e., ax is of vector space  
n
R and on right 
aT(x) is scalar multiplication of 
m
R . 
 
Value Addition: Do you know: 
Two axioms in the linear transformations may be written in one condition 
only i.e. 
Definition: A mapping :
nm
T R R ? is called the linear transformation if it 
satisfies the following condition 
( ) ( ) ( ), , ,
n
T ax y aT x bT y x y R and a b R ? ? ? ? ? ? ? . 
Page 4


Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Algebra-I 
Lesson: Linear Transformations  
Lesson Developer: Chaman Singh  
College/Department: Acharya Narendra Dev College (D.U.)  
 
 
 
 
 
 
 
 
 
 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 2 
 
 
 
Table of Contents: 
 Chapter : Linear Transformations 
? 1. Learning Outcomes 
? 2. Linear Transformation 
? 3. Matrix Transformation 
o 3.1. Contraction and Dilation 
o 3.2. Shear Transformation 
? 4. Matrix of a Linear Transformation 
o 4.1. Basis 
o 4.2. Steps to find the matrix of the transformation  
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
1. Learning Outcomes: 
After studying the whole contents of this chapter, students will be able to 
understand:  
? Linear transformation 
? Matrix transformation 
? Contraction and Dilation 
? Shear Transformation 
? Matrix of linear transformations 
? Basis of linear transformations 
? How to find the matrix of a linear transformation 
 
 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 3 
 
 
 
2. Linear Transformation: 
Consider the Euclidean spaces 
nm
R and R . A mapping :
nm
T R R ?  is called 
the linear mapping or linear transformation if T satisfies the following two 
axioms: 
 
( ) ( ) ( ) ( )
( ) ( ) ( )
I T x y T x T y
II T ax aT x
? ? ?
?
   
Where all ,
n
x y R ?  and all aR ? . 
Value Addition: Note 
The set 
n
R is said to be the domain of T, and 
m
R co-domain of T. 
 
Value Additions: Do you know? 
A mapping or function from X to Y i.e. : f X Y ?  is a rule that assigns to 
each element x in X a unique element y in Y. The set X is called the 
domain of f and the set Y is called the co-domain of f. For , xX ? the 
element yY ? , such that f(x) = y, is called the image of x (under the 
action of f) and the set of all images yY ? is called the range of f. 
 
Value Addition: Remember 
Note: a mapping :
nm
T R R ? is linear if it "preserves" the two basic 
operations of a vector space, 
(i) vector addition, i.e., ( ) ( ) ( ), , ( ), ( )
nm
T x y T x T y x y R andT x T y R ? ? ? ? ? ? , 
addition of x and y on left is the addition of vectors of 
n
R and addition of 
T(x) and T(y) on right is addition of vectors of 
m
R . 
(ii) scalar multiplication, i.e.,  
( ) ( ),
n
T ax aT x x R and a R ? ? ? ? ?  
Scalar multiplication on left i.e., ax is of vector space  
n
R and on right 
aT(x) is scalar multiplication of 
m
R . 
 
Value Addition: Do you know: 
Two axioms in the linear transformations may be written in one condition 
only i.e. 
Definition: A mapping :
nm
T R R ? is called the linear transformation if it 
satisfies the following condition 
( ) ( ) ( ), , ,
n
T ax y aT x bT y x y R and a b R ? ? ? ? ? ? ? . 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 4 
 
 
Example 1:  consider the mapping 
32
: T R R ? be defined as  
  
3
1 2 3 1 3 1 2 3
( , , ) ( ,0, ), ( , , ) T x x x x x x x x R ? ? ?  
Check whether T is linear or not. 
Solution: let 
1 2 3 1 2 3
( , , ) ( , , ) X x x x andY y y y ?? be any two vectors of 
3
R 
(i) vector addition :  
  
1 2 3 1 2 3 1 1 2 2 3 3
( , , ) ( , , ) ( , , ) X Y x x x y y y x y x y x y ? ? ? ? ? ? ?  
By definition of T, we have 
 
1 2 3 1 3 1 2 3 1 3
( , , ) ( ,0, ) ( , , ) ( ,0, ) T x x x x x and T y y y y y ??  
Now,  
 
1 1 2 2 3 3
1 1 3 3
1 3 1 3
1 2 3 1 2 3
3
( ) ( , , )
( ,0, )
( ,0, ) ( ,0, )
( , , ) ( , , )
( ) ( )
,
T X Y T x y x y x y
x y x y
x x y y
T x x x T y y y
T X T Y
X Y R
? ? ? ? ?
? ? ?
??
??
??
??
  
(ii) scalar multiplication: For any scalar 
3
1 2 3
( , , ) a Randvector X x x x R ? ? ? . 
 
1 2 3 1 2 3
( , , ) ( , , ) aX a x x x ax ax ax ??   
Now, 
 
1 2 3
13
13
( ) ( , , )
( ,0, )
( ,0, )
()
T aX T ax ax ax
ax ax
a x x
aT X
?
?
?
?
  
Thus T satisfies both the axioms therefore T is a linear transformation. 
Value Addition: Note 
Axiom (i) say that the result T(X + Y) of first adding X and Y in 
n
R and 
then applying T is same as first applying T to X and Y and then adding 
T(X) and T(Y) in 
m
R . 
 
Page 5


Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Algebra-I 
Lesson: Linear Transformations  
Lesson Developer: Chaman Singh  
College/Department: Acharya Narendra Dev College (D.U.)  
 
 
 
 
 
 
 
 
 
 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 2 
 
 
 
Table of Contents: 
 Chapter : Linear Transformations 
? 1. Learning Outcomes 
? 2. Linear Transformation 
? 3. Matrix Transformation 
o 3.1. Contraction and Dilation 
o 3.2. Shear Transformation 
? 4. Matrix of a Linear Transformation 
o 4.1. Basis 
o 4.2. Steps to find the matrix of the transformation  
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
1. Learning Outcomes: 
After studying the whole contents of this chapter, students will be able to 
understand:  
? Linear transformation 
? Matrix transformation 
? Contraction and Dilation 
? Shear Transformation 
? Matrix of linear transformations 
? Basis of linear transformations 
? How to find the matrix of a linear transformation 
 
 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 3 
 
 
 
2. Linear Transformation: 
Consider the Euclidean spaces 
nm
R and R . A mapping :
nm
T R R ?  is called 
the linear mapping or linear transformation if T satisfies the following two 
axioms: 
 
( ) ( ) ( ) ( )
( ) ( ) ( )
I T x y T x T y
II T ax aT x
? ? ?
?
   
Where all ,
n
x y R ?  and all aR ? . 
Value Addition: Note 
The set 
n
R is said to be the domain of T, and 
m
R co-domain of T. 
 
Value Additions: Do you know? 
A mapping or function from X to Y i.e. : f X Y ?  is a rule that assigns to 
each element x in X a unique element y in Y. The set X is called the 
domain of f and the set Y is called the co-domain of f. For , xX ? the 
element yY ? , such that f(x) = y, is called the image of x (under the 
action of f) and the set of all images yY ? is called the range of f. 
 
Value Addition: Remember 
Note: a mapping :
nm
T R R ? is linear if it "preserves" the two basic 
operations of a vector space, 
(i) vector addition, i.e., ( ) ( ) ( ), , ( ), ( )
nm
T x y T x T y x y R andT x T y R ? ? ? ? ? ? , 
addition of x and y on left is the addition of vectors of 
n
R and addition of 
T(x) and T(y) on right is addition of vectors of 
m
R . 
(ii) scalar multiplication, i.e.,  
( ) ( ),
n
T ax aT x x R and a R ? ? ? ? ?  
Scalar multiplication on left i.e., ax is of vector space  
n
R and on right 
aT(x) is scalar multiplication of 
m
R . 
 
Value Addition: Do you know: 
Two axioms in the linear transformations may be written in one condition 
only i.e. 
Definition: A mapping :
nm
T R R ? is called the linear transformation if it 
satisfies the following condition 
( ) ( ) ( ), , ,
n
T ax y aT x bT y x y R and a b R ? ? ? ? ? ? ? . 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 4 
 
 
Example 1:  consider the mapping 
32
: T R R ? be defined as  
  
3
1 2 3 1 3 1 2 3
( , , ) ( ,0, ), ( , , ) T x x x x x x x x R ? ? ?  
Check whether T is linear or not. 
Solution: let 
1 2 3 1 2 3
( , , ) ( , , ) X x x x andY y y y ?? be any two vectors of 
3
R 
(i) vector addition :  
  
1 2 3 1 2 3 1 1 2 2 3 3
( , , ) ( , , ) ( , , ) X Y x x x y y y x y x y x y ? ? ? ? ? ? ?  
By definition of T, we have 
 
1 2 3 1 3 1 2 3 1 3
( , , ) ( ,0, ) ( , , ) ( ,0, ) T x x x x x and T y y y y y ??  
Now,  
 
1 1 2 2 3 3
1 1 3 3
1 3 1 3
1 2 3 1 2 3
3
( ) ( , , )
( ,0, )
( ,0, ) ( ,0, )
( , , ) ( , , )
( ) ( )
,
T X Y T x y x y x y
x y x y
x x y y
T x x x T y y y
T X T Y
X Y R
? ? ? ? ?
? ? ?
??
??
??
??
  
(ii) scalar multiplication: For any scalar 
3
1 2 3
( , , ) a Randvector X x x x R ? ? ? . 
 
1 2 3 1 2 3
( , , ) ( , , ) aX a x x x ax ax ax ??   
Now, 
 
1 2 3
13
13
( ) ( , , )
( ,0, )
( ,0, )
()
T aX T ax ax ax
ax ax
a x x
aT X
?
?
?
?
  
Thus T satisfies both the axioms therefore T is a linear transformation. 
Value Addition: Note 
Axiom (i) say that the result T(X + Y) of first adding X and Y in 
n
R and 
then applying T is same as first applying T to X and Y and then adding 
T(X) and T(Y) in 
m
R . 
 
Linear Transformations 
Institute of Lifelong Learning                                                                                                                                       pg. 5 
 
Alternative method: instead of satisfying the two axioms, we will satisfy 
the single axiom, i.e., for all ,, a b R ? we have 
 
1 2 3 1 2 3
1 2 3 1 2 3
1 1 2 2 3 3
( , , ) ( , , )
( , , ) ( , , )
( , , )
aX bY a x x x b y y y
ax ax ax by by by
aX bY ax by ax by ax by
? ? ?
??
? ? ? ? ?
  
Thus, 
 
1 1 2 2 3 3
1 1 3 3
1 3 1 3
1 3 1 3
( ) ( , , )
( ,0, )
( ,0, ) ( ,0, )
( ,0, ) ( ,0, )
( ) ( )
T aX bY T ax by ax by ax by
ax by ax by
ax ax by by
a x x b y y
aT X bT Y
? ? ? ? ?
? ? ?
??
??
??
  
Thus, we have ( ) ( ) ( ) T aX bY aT X bT Y ? ? ?  
Hence, T is a linear transformation. 
Value Addition: Do you know? 
Every linear transformation maps the zero vector of 
n
R onto the zero 
vector of 
m
R . I.e., (0) 0 T ? . 
 
Example 2: consider the mapping 
23
: T R R ? defined by 
  
1 2 1 2 1 2 1 2
( , ) ( 1,2 , 3 ) T x x x x x x x x ? ? ? ? ?  
Check whether T is linear or not. 
Solution: Let 
1 2 1 2
( , ) ( , ) X x x and Y y y ?? be any two vectors of 
2
R . By the 
definition of the mapping, we have  
12
1 2 1 2 1 2
( ) ( , )
( 1,2 , 3 )
T X T x x
x x x x x x
?
? ? ? ? ?
   (1) 
And 
1 2 1 3
1 2 1 2 1 2
( ) ( , ) ( ,0, )
( 1,2 , 3 )
T Y T y y y y
y y y y y y
??
? ? ? ? ?
   (2)

	Algebra- Engineering Maths 9 docs

Algebra- Engineering Maths

9 docs

Join Course for Free

FAQs on Lecture 7 - Linear Transformations - Algebra- Engineering Maths - Engineering Mathematics

1. What is a linear transformation in engineering mathematics?

Ans. In engineering mathematics, a linear transformation refers to a function that maps vectors from one vector space to another in a way that preserves the linear structure. It follows the properties of linearity, such as preserving addition and scalar multiplication.

2. How are linear transformations represented mathematically?

Ans. Linear transformations in engineering mathematics are often represented using matrices. Each column of the matrix represents the image of the corresponding basis vector under the transformation. The matrix multiplication of the transformation matrix with a vector gives the resulting transformed vector.

3. What are the applications of linear transformations in engineering?

Ans. Linear transformations have various applications in engineering. Some examples include image and signal processing, control systems, electrical circuit analysis, and computer graphics. They are used to model and analyze systems, perform transformations on data, and solve engineering problems.

4. Can a linear transformation change the dimension of a vector space?

Ans. No, a linear transformation cannot change the dimension of a vector space. The dimension of the vector space remains the same before and after the transformation. However, the linear transformation may map vectors from higher-dimensional spaces to lower-dimensional spaces.

5. How can we determine if a transformation is linear or not?

Ans. To determine if a transformation is linear, we need to check two properties: preservation of addition and preservation of scalar multiplication. If the transformation satisfies these properties, it is linear. Mathematically, a transformation T is linear if T(u + v) = T(u) + T(v) and T(cu) = cT(u), where u and v are vectors and c is a scalar.

Related Exams

Engineering Mathematics

About this Document

	4.72/5 Rating
	Nov 23, 2024 Last updated

Document Description: Lecture 7 - Linear Transformations for Engineering Mathematics 2024 is part of Algebra- Engineering Maths preparation. The notes and questions for Lecture 7 - Linear Transformations have been prepared according to the Engineering Mathematics exam syllabus. Information about Lecture 7 - Linear Transformations covers topics like and Lecture 7 - Linear Transformations Example, for Engineering Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Lecture 7 - Linear Transformations.

Introduction of Lecture 7 - Linear Transformations in English is available as part of our Algebra- Engineering Maths for Engineering Mathematics & Lecture 7 - Linear Transformations in Hindi for Algebra- Engineering Maths course. Download more important topics related with notes, lectures and mock test series for Engineering Mathematics Exam by signing up for free. Engineering Mathematics: Lecture 7 - Linear Transformations | Algebra- Engineering Maths - Engineering Mathematics

Description

Full syllabus notes, lecture & questions for Lecture 7 - Linear Transformations | Algebra- Engineering Maths - Engineering Mathematics - Engineering Mathematics | Plus excerises question with solution to help you revise complete syllabus for Algebra- Engineering Maths | Best notes, free PDF download

Information about Lecture 7 - Linear Transformations

In this doc you can find the meaning of Lecture 7 - Linear Transformations defined & explained in the simplest way possible. Besides explaining types of Lecture 7 - Linear Transformations theory, EduRev gives you an ample number of questions to practice Lecture 7 - Linear Transformations tests, examples and also practice Engineering Mathematics tests

	Algebra- Engineering Maths 9 docs

Algebra- Engineering Maths

9 docs

Join Course for Free

Download as PDF

Explore Courses for Engineering Mathematics exam

Signup for Free!

Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.

Start learning for Free

10M+ students study on EduRev

Important questions

ppt

Objective type Questions

Exam

video lectures

Extra Questions

mock tests for examination

practice quizzes

Sample Paper

Lecture 7 - Linear Transformations | Algebra- Engineering Maths - Engineering Mathematics

shortcuts and tricks

Lecture 7 - Linear Transformations | Algebra- Engineering Maths - Engineering Mathematics

pdf

MCQs

Previous Year Questions with Solutions

Summary

Viva Questions

study material

past year papers

Semester Notes

Free

Lecture 7 - Linear Transformations | Algebra- Engineering Maths - Engineering Mathematics

;

Additional Information about Lecture 7 - Linear Transformations for Engineering Mathematics Preparation

Lecture 7 - Linear Transformations Free PDF Download

The Lecture 7 - Linear Transformations is an invaluable resource that delves deep into the core of the Engineering Mathematics exam. These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective. With the help of these notes, you can grasp complex subjects quickly, revise important points easily, and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner, allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material, or toppers' notes, this PDF has got you covered. Download the Lecture 7 - Linear Transformations now and kickstart your journey towards success in the Engineering Mathematics exam.

Importance of Lecture 7 - Linear Transformations

The importance of Lecture 7 - Linear Transformations cannot be overstated, especially for Engineering Mathematics aspirants. This document holds the key to success in the Engineering Mathematics exam. It offers a detailed understanding of the concept, providing invaluable insights into the topic. By knowing the concepts well in advance, students can plan their preparation effectively. Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.

Lecture 7 - Linear Transformations Notes

Lecture 7 - Linear Transformations Notes offer in-depth insights into the specific topic to help you master it with ease. This comprehensive document covers all aspects related to Lecture 7 - Linear Transformations. It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation. Practice papers and question papers enable you to assess your progress effectively. Additionally, the paper analysis provides valuable tips for tackling the exam strategically. Access to Toppers' notes gives you an edge in understanding complex concepts. Whether you're a beginner or aiming for advanced proficiency, Lecture 7 - Linear Transformations Notes on EduRev are your ultimate resource for success.

Lecture 7 - Linear Transformations Engineering Mathematics Questions

The "Lecture 7 - Linear Transformations Engineering Mathematics Questions" guide is a valuable resource for all aspiring students preparing for the Engineering Mathematics exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation. The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level. Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.

Study Lecture 7 - Linear Transformations on the App

Students of Engineering Mathematics can study Lecture 7 - Linear Transformations alongwith tests & analysis from the EduRev app, which will help them while preparing for their exam. Apart from the Lecture 7 - Linear Transformations, students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc. The EduRev App will make your learning easier as you can access it from anywhere you want. The content of Lecture 7 - Linear Transformations is prepared as per the latest Engineering Mathematics syllabus.

Education Revolution