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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)

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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.

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