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Page 1
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Dual Vector Spaces
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Page 2
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Dual Vector Spaces
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of Linear Functional and Dual Vector Space
4. Dual Basis for a Finite Dimensional Vector Space
5. Bi-dual of a Vector Space
6. Annihilator Subspace
7. Summary
8. Exercises
9. Glossary and Further Reading
10. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a linear functional on a vector space.
? explain the concept of the dual of a vector space over a field.
? state the meaning of the dual basis of a vector space.
? compute the dual basis for any given basis for a vector space.
? define the concept of bi-dual of a vector space.
? explain the notion of annihilator subspace of the dual vector
space.
Page 3
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Dual Vector Spaces
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of Linear Functional and Dual Vector Space
4. Dual Basis for a Finite Dimensional Vector Space
5. Bi-dual of a Vector Space
6. Annihilator Subspace
7. Summary
8. Exercises
9. Glossary and Further Reading
10. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a linear functional on a vector space.
? explain the concept of the dual of a vector space over a field.
? state the meaning of the dual basis of a vector space.
? compute the dual basis for any given basis for a vector space.
? define the concept of bi-dual of a vector space.
? explain the notion of annihilator subspace of the dual vector
space.
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
One of the keys to understand the structure of a vector space is to learn more about the
linear transformations on that vector space. Knowledge about any vector space is greatly
enhanced if we study the linear transformations of that vector space into the underlying
field. This is the main agenda for this lesson. The principal theme of this unit is: “scalar-
valued linear transformations of a finite dimensional vector space”. In other words, we are
going to learn about such linear transformations ?? :?? ? F, where ?? is a finite dimensional
vector space over the field F. These linear transformations are also known as linear
functionals. The set of all linear functionals on a vector space ?? constitutes what is known
as the dual vector space of ?? .
Let us start our unit with these definitions.
3. Definition of Linear Functional and Dual Vector Space:
Suppose ?? and ?? are two vector spaces over the same field F. Then, we know that the set
?? ?? ?? F
?? ,?? = ?? :?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is a vector space over the field F under the natural point wise addition and scalar
multiplication. So, in particular, since the field F itself is a vector space over F, the set
?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is also a vector space over the field F under natural addition and scalar multiplication. It is
this vector space which is known as the dual of the vector space ?? . So, let us now make the
desired definition.
Definition of Dual of a Vector Space: Let ?? be a vector space over a field F. The
set
?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is called the dual vector space of ?? . It is denoted by ?? *
. Each member of the dual vector
space is called a linear functional on ?? .
Before going any further, let us look at some examples of linear functionals.
Page 4
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Dual Vector Spaces
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of Linear Functional and Dual Vector Space
4. Dual Basis for a Finite Dimensional Vector Space
5. Bi-dual of a Vector Space
6. Annihilator Subspace
7. Summary
8. Exercises
9. Glossary and Further Reading
10. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a linear functional on a vector space.
? explain the concept of the dual of a vector space over a field.
? state the meaning of the dual basis of a vector space.
? compute the dual basis for any given basis for a vector space.
? define the concept of bi-dual of a vector space.
? explain the notion of annihilator subspace of the dual vector
space.
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
One of the keys to understand the structure of a vector space is to learn more about the
linear transformations on that vector space. Knowledge about any vector space is greatly
enhanced if we study the linear transformations of that vector space into the underlying
field. This is the main agenda for this lesson. The principal theme of this unit is: “scalar-
valued linear transformations of a finite dimensional vector space”. In other words, we are
going to learn about such linear transformations ?? :?? ? F, where ?? is a finite dimensional
vector space over the field F. These linear transformations are also known as linear
functionals. The set of all linear functionals on a vector space ?? constitutes what is known
as the dual vector space of ?? .
Let us start our unit with these definitions.
3. Definition of Linear Functional and Dual Vector Space:
Suppose ?? and ?? are two vector spaces over the same field F. Then, we know that the set
?? ?? ?? F
?? ,?? = ?? :?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is a vector space over the field F under the natural point wise addition and scalar
multiplication. So, in particular, since the field F itself is a vector space over F, the set
?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is also a vector space over the field F under natural addition and scalar multiplication. It is
this vector space which is known as the dual of the vector space ?? . So, let us now make the
desired definition.
Definition of Dual of a Vector Space: Let ?? be a vector space over a field F. The
set
?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is called the dual vector space of ?? . It is denoted by ?? *
. Each member of the dual vector
space is called a linear functional on ?? .
Before going any further, let us look at some examples of linear functionals.
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 4
3.1 Examples of Linear Functionals on a vector space:
(1) Let us take R
?? as a vector space over R. The map
?? 1
,?? 2
,… ,?? ?? ,… ,?? ?? ? ?? ?? for ?? = 1 1 ??
defines a linear transformation from R
?? into R. Clearly, therefore, this is a linear
functional on R
?? . This linear functional is also known as the ?? ?? h
projection map on
R
?? .
(2) Taking the vector space ?? ?? ×?? (R) of all square matrices of order ?? over
the field of real numbers, we see that the trace map
?? ?? ?? ?
1
ii
n
i
a
?
?
defines a linear transformation from ?? ?? ×?? (R) onto R. Therefore, this is a linear
functional on ?? ?? ×?? (R).
(3) Let P
?? R = {?? :R ? R: ?? ?? = ?? 0
+ ?? 1
?? +? +?? ?? ?? ?? ?? ?? ?? ?? ?? ?? R ? ?? = 0 1 ?? }
be the vector space of all polynomials of degree at most ?? . The mapping
?? ? ?? ?? ?? 1
0
gives a linear mapping of P
?? R into R and is, therefore, an example of a linear
functional on P
?? R .
I.Q.1
4. Dual Basis for a Finite Dimensional Vector Space:
If ?? and ?? are finite dimensional vector spaces over the field F with ?? ?? ?? F
?? = ?? and
?? ?? ?? F
?? = ?? , then ?? ?? ?? F
?? ,?? is also a finite dimensional vector spaces over the field F with
?? ?? ?? F
?? ?? ?? F
?? ,?? = ?? ?? ?? F
?? .?? ?? ?? F
?? = ?? ?? .
In particular, for any ?? -dimensional vector space ?? over the field F, the set
?? *
= ?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is a finite dimensional vector space over the field F with
?? ?? ?? F
?? *
= ?? ?? ?? F
?? ?? ?? F
?? ,F = ?? ?? ?? F
?? . 1 = ?? ?? ?? F
?? = ?? .
Thus, ?? *
has a basis of ?? elements. We shall now construct a basis for the dual space ?? *
.
Theorem 1 (Dual Basis): Let ?? be an ?? -dimensional vector space over a field F having
the set {?? 1
,?? 2
,… ,?? ?? } as a basis. For each ?? = 1 1 ?? , let ?? ?? :?? ? F be a linear transformation
such that
Page 5
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Dual Vector Spaces
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of Linear Functional and Dual Vector Space
4. Dual Basis for a Finite Dimensional Vector Space
5. Bi-dual of a Vector Space
6. Annihilator Subspace
7. Summary
8. Exercises
9. Glossary and Further Reading
10. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a linear functional on a vector space.
? explain the concept of the dual of a vector space over a field.
? state the meaning of the dual basis of a vector space.
? compute the dual basis for any given basis for a vector space.
? define the concept of bi-dual of a vector space.
? explain the notion of annihilator subspace of the dual vector
space.
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
One of the keys to understand the structure of a vector space is to learn more about the
linear transformations on that vector space. Knowledge about any vector space is greatly
enhanced if we study the linear transformations of that vector space into the underlying
field. This is the main agenda for this lesson. The principal theme of this unit is: “scalar-
valued linear transformations of a finite dimensional vector space”. In other words, we are
going to learn about such linear transformations ?? :?? ? F, where ?? is a finite dimensional
vector space over the field F. These linear transformations are also known as linear
functionals. The set of all linear functionals on a vector space ?? constitutes what is known
as the dual vector space of ?? .
Let us start our unit with these definitions.
3. Definition of Linear Functional and Dual Vector Space:
Suppose ?? and ?? are two vector spaces over the same field F. Then, we know that the set
?? ?? ?? F
?? ,?? = ?? :?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is a vector space over the field F under the natural point wise addition and scalar
multiplication. So, in particular, since the field F itself is a vector space over F, the set
?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is also a vector space over the field F under natural addition and scalar multiplication. It is
this vector space which is known as the dual of the vector space ?? . So, let us now make the
desired definition.
Definition of Dual of a Vector Space: Let ?? be a vector space over a field F. The
set
?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is called the dual vector space of ?? . It is denoted by ?? *
. Each member of the dual vector
space is called a linear functional on ?? .
Before going any further, let us look at some examples of linear functionals.
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 4
3.1 Examples of Linear Functionals on a vector space:
(1) Let us take R
?? as a vector space over R. The map
?? 1
,?? 2
,… ,?? ?? ,… ,?? ?? ? ?? ?? for ?? = 1 1 ??
defines a linear transformation from R
?? into R. Clearly, therefore, this is a linear
functional on R
?? . This linear functional is also known as the ?? ?? h
projection map on
R
?? .
(2) Taking the vector space ?? ?? ×?? (R) of all square matrices of order ?? over
the field of real numbers, we see that the trace map
?? ?? ?? ?
1
ii
n
i
a
?
?
defines a linear transformation from ?? ?? ×?? (R) onto R. Therefore, this is a linear
functional on ?? ?? ×?? (R).
(3) Let P
?? R = {?? :R ? R: ?? ?? = ?? 0
+ ?? 1
?? +? +?? ?? ?? ?? ?? ?? ?? ?? ?? ?? R ? ?? = 0 1 ?? }
be the vector space of all polynomials of degree at most ?? . The mapping
?? ? ?? ?? ?? 1
0
gives a linear mapping of P
?? R into R and is, therefore, an example of a linear
functional on P
?? R .
I.Q.1
4. Dual Basis for a Finite Dimensional Vector Space:
If ?? and ?? are finite dimensional vector spaces over the field F with ?? ?? ?? F
?? = ?? and
?? ?? ?? F
?? = ?? , then ?? ?? ?? F
?? ,?? is also a finite dimensional vector spaces over the field F with
?? ?? ?? F
?? ?? ?? F
?? ,?? = ?? ?? ?? F
?? .?? ?? ?? F
?? = ?? ?? .
In particular, for any ?? -dimensional vector space ?? over the field F, the set
?? *
= ?? ?? ?? F
?? ,F = ?? :?? ? F ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? }
is a finite dimensional vector space over the field F with
?? ?? ?? F
?? *
= ?? ?? ?? F
?? ?? ?? F
?? ,F = ?? ?? ?? F
?? . 1 = ?? ?? ?? F
?? = ?? .
Thus, ?? *
has a basis of ?? elements. We shall now construct a basis for the dual space ?? *
.
Theorem 1 (Dual Basis): Let ?? be an ?? -dimensional vector space over a field F having
the set {?? 1
,?? 2
,… ,?? ?? } as a basis. For each ?? = 1 1 ?? , let ?? ?? :?? ? F be a linear transformation
such that
Dual Vector Spaces
Institute of Lifelong Learning, University of Delhi pg. 5
?? ?? ?? ?? =
1,
0,
ij
ij
? ?
?
?
?
? ?? ,?? = 1 1 ?? .
Then, the set {?? 1
,?? 2
,… ,?? ?? } is a basis for the dual vector space ?? *
.
Proof: Let ?? = {?? 1
,?? 2
,… ,?? ?? } and ?? *
= {?? 1
,?? 2
,… ,?? ?? }. Clearly, we see by the definition of ?? ?? ’s that
?? *
? ?? *
. Now, the proof amounts to showing that
(1) ?? *
is a linearly independent set (over the field F) in ?? *
; and
(2) ?? ?? *
= ?? *
.
We first have a proof for (1).
Suppose that
1
n
i
ii
f ?
?
?
= 0
for any scalars ?? ?? ?? F. Then, for each basis element ?? ?? of ?? , we see that
1
n
i
ii
f ?
?
?
?? ?? = 0
? ?? ?? ?? ?? ?? ?? = 0 (Because ?? ?? ?? ?? = 0, whenever, ?? ? ?? .)
? ?? ?? = 0 (? ?? = 1 1 ?? .)
and therefore, ?? *
is a linearly independent set (over the field F) in ?? *
.
This proves (1).
Next we have a proof for (2).
First we observe that any ?? ?? ?? can be represented as
?? =
1
n
i
ii
v ?
?
?
. (This is because ?? is a basis for ?? .)
Now, we see that for each ?? = 1 1 ?? ,
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FAQs on Lecture 1 - Dual Vector Spaces - Linear Algebra - Engineering Mathematics
| 1. What is a dual vector space? |  |
Ans. A dual vector space is the set of all linear functionals on a given vector space. It consists of all possible linear maps from the vector space to its underlying field of scalars.
| 2. How is a dual vector space related to engineering mathematics? | |
Ans. In engineering mathematics, the concept of a dual vector space is often used to analyze and solve problems involving vector spaces. It allows for the representation of vectors as linear functionals and provides a framework for understanding various mathematical operations in engineering applications.
| 3. What is the importance of dual vector spaces in engineering mathematics? | |
Ans. Dual vector spaces are important in engineering mathematics as they provide a way to define and manipulate vectors in a more abstract and general manner. This abstraction allows engineers to apply mathematical concepts and techniques to a wide range of engineering problems, such as optimization, control systems, and signal processing.
| 4. How do you determine the dimension of a dual vector space? | |
Ans. The dimension of a dual vector space is equal to the dimension of the original vector space. This is because for every basis vector in the original space, there exists a corresponding dual basis vector in the dual space. Thus, the number of basis vectors in both spaces is the same, resulting in equal dimensions.
| 5. Can a vector space and its dual vector space have different dimensions? | |
Ans. No, a vector space and its dual vector space always have the same dimension. This is a fundamental property of dual spaces. If the original vector space has a finite dimension of n, then its dual space will also have a finite dimension of n.
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