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Lecture 6 - Linear Transformations
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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
??
? ? ? ?
??
Read MoreFAQs on Lecture 6 - Linear Transformations
| 1. What exactly is a linear transformation and how does it differ from a regular function? |  |
Ans. A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. Unlike regular functions, linear transformations must satisfy two properties: T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors and scalars. This special structure makes them fundamental in engineering mathematics and applications involving matrix operations.
| 2. How do I know if a transformation is linear or non-linear in engineering problems? | |
Ans. Check two conditions: additivity (T(u + v) = T(u) + T(v)) and homogeneity (T(cu) = cT(u)). If both hold, the transformation is linear. Non-linear transformations fail at least one condition-for example, T(x) = x² or T(x) = x + 1. Visualising transformations using mind maps helps identify these properties quickly during exams.
| 3. What's the connection between linear transformations and matrices in CBSE engineering mathematics? | |
Ans. Every linear transformation between finite-dimensional vector spaces can be represented as a matrix. The matrix columns contain images of basis vectors under the transformation. Conversely, every matrix defines a linear transformation. This correspondence makes matrices powerful tools for computing transformations, solving systems, and analysing geometric effects like rotations and reflections.
| 4. Why do eigenvalues and eigenvectors matter when studying linear transformations? | |
Ans. Eigenvalues and eigenvectors reveal how linear transformations behave. An eigenvector remains in the same direction after transformation, scaled by its eigenvalue. Finding these special pairs simplifies understanding transformation effects, diagonalises matrices, and solves differential equations. They're critical for analysing stability in engineering applications and optimisation problems.
| 5. How can I use the kernel and image of a linear transformation to understand its properties? | |
Ans. The kernel (null space) contains all vectors mapping to zero; the image (range) contains all possible outputs. Their dimensions-nullity and rank-satisfy the rank-nullity theorem: rank + nullity = domain dimension. These concepts reveal whether a transformation is injective, surjective, or bijective, essential for determining invertibility and solution existence in linear systems.
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