Lecture 7 - Vector Spaces | Linear Algebra - Engineering Mathematics PDF Download

 Page 1


Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Spaces 
Paper: Linear Algebra 
Lesson Developer: Pushpendra Kumar Vashishtha and 
Dr. Arvind 
College/Department: Kamala Nehru College (D.U) / 
Hansraj College (D.U), University of Delhi 
 
 
 
 
 
 
  
Page 2


Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Spaces 
Paper: Linear Algebra 
Lesson Developer: Pushpendra Kumar Vashishtha and 
Dr. Arvind 
College/Department: Kamala Nehru College (D.U) / 
Hansraj College (D.U), University of Delhi 
 
 
 
 
 
 
  
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of Contents:    
 
Chapter : Vector Spaces                                                   
? Learning Outcomes 
? Introduction  
? Preliminaries 
? Vector Spaces 
? Axioms 
? Examples 
? Properties 
? Subspaces 
? Linear combination 
? Linear Span 
? Row spaces 
? Summary  
? Multiple Choice Questions 
? References 
 
 
 
1. Learning Outcomes: 
After taking a visit of this chapter the reader will be able to learn: 
? Set and Binary operations 
? Algebraic Structure 
? Vector spaces 
? Sub spaces 
? Linear Span 
? Row Spaces 
 
 
 
Page 3


Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Spaces 
Paper: Linear Algebra 
Lesson Developer: Pushpendra Kumar Vashishtha and 
Dr. Arvind 
College/Department: Kamala Nehru College (D.U) / 
Hansraj College (D.U), University of Delhi 
 
 
 
 
 
 
  
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of Contents:    
 
Chapter : Vector Spaces                                                   
? Learning Outcomes 
? Introduction  
? Preliminaries 
? Vector Spaces 
? Axioms 
? Examples 
? Properties 
? Subspaces 
? Linear combination 
? Linear Span 
? Row spaces 
? Summary  
? Multiple Choice Questions 
? References 
 
 
 
1. Learning Outcomes: 
After taking a visit of this chapter the reader will be able to learn: 
? Set and Binary operations 
? Algebraic Structure 
? Vector spaces 
? Sub spaces 
? Linear Span 
? Row Spaces 
 
 
 
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
2. Introduction:  
 Mathematics is the game of numbers. What if there is no number to 
play with? O.K then we will be choosing some objects, as we are keenly 
interested in playing the game. This is reason why Mathematics has been 
divided in two parts,  One in which we do some calculations and is more 
realistic so called Applied Mathematics , other in which the things or the 
objects does not appear and we play an abstract game , is called as Pure 
Mathematics. Pure mathematics is a tree and Linear algebra is one of its 
branches. In Linear Algebra we take some objects and then spread them 
by applying some operations to make some other new objects and 
continue this process until we get a family of objects which is complete in 
itself and further cannot be expanded. The expansion of objects is done 
linearly in form of linear combination that’s why this branch is called 
Linear Algebra.  
 In this chapter we will be discussing the foundation of Linear 
Algebra so called Vector Spaces. It is hard to overstate the importance of 
the idea of a vector space, a concept which has found application in the 
areas of mathematics, engineering, physics, chemistry, biology, the social 
sciences and others.  
3. Preliminaries: 
 Before going on war we need weapons. you are here to fight with 
vectors in their homes named as “Vector Spaces” , so we first learn some 
of the basic concepts that are very essential for learning Vector Spaces. 
We will not be going into depth as it is assumed that the reader is familiar 
with these concepts. 
3.1. Set : A set is well defined collection of objects. The term well defined 
specifies that there must be some rule with the help of which we can 
unambiguously say that the particular element belongs to that set or not.  
? Sets are always denoted by the capital letters and the elements are 
denoted by the small letters. 
? The elements of a set are written in the braces {} separated by 
commas. 
Page 4


Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Spaces 
Paper: Linear Algebra 
Lesson Developer: Pushpendra Kumar Vashishtha and 
Dr. Arvind 
College/Department: Kamala Nehru College (D.U) / 
Hansraj College (D.U), University of Delhi 
 
 
 
 
 
 
  
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of Contents:    
 
Chapter : Vector Spaces                                                   
? Learning Outcomes 
? Introduction  
? Preliminaries 
? Vector Spaces 
? Axioms 
? Examples 
? Properties 
? Subspaces 
? Linear combination 
? Linear Span 
? Row spaces 
? Summary  
? Multiple Choice Questions 
? References 
 
 
 
1. Learning Outcomes: 
After taking a visit of this chapter the reader will be able to learn: 
? Set and Binary operations 
? Algebraic Structure 
? Vector spaces 
? Sub spaces 
? Linear Span 
? Row Spaces 
 
 
 
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
2. Introduction:  
 Mathematics is the game of numbers. What if there is no number to 
play with? O.K then we will be choosing some objects, as we are keenly 
interested in playing the game. This is reason why Mathematics has been 
divided in two parts,  One in which we do some calculations and is more 
realistic so called Applied Mathematics , other in which the things or the 
objects does not appear and we play an abstract game , is called as Pure 
Mathematics. Pure mathematics is a tree and Linear algebra is one of its 
branches. In Linear Algebra we take some objects and then spread them 
by applying some operations to make some other new objects and 
continue this process until we get a family of objects which is complete in 
itself and further cannot be expanded. The expansion of objects is done 
linearly in form of linear combination that’s why this branch is called 
Linear Algebra.  
 In this chapter we will be discussing the foundation of Linear 
Algebra so called Vector Spaces. It is hard to overstate the importance of 
the idea of a vector space, a concept which has found application in the 
areas of mathematics, engineering, physics, chemistry, biology, the social 
sciences and others.  
3. Preliminaries: 
 Before going on war we need weapons. you are here to fight with 
vectors in their homes named as “Vector Spaces” , so we first learn some 
of the basic concepts that are very essential for learning Vector Spaces. 
We will not be going into depth as it is assumed that the reader is familiar 
with these concepts. 
3.1. Set : A set is well defined collection of objects. The term well defined 
specifies that there must be some rule with the help of which we can 
unambiguously say that the particular element belongs to that set or not.  
? Sets are always denoted by the capital letters and the elements are 
denoted by the small letters. 
? The elements of a set are written in the braces {} separated by 
commas. 
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
For Example: Set of Natural Numbers, N={1,2,3,4,. . . .} 
For Example: Set of Integers Z={. . . ,-3,-2,-1,0,1,2,3, . . .} 
3.2. Mapping: A mapping, between two non empty sets A and B, is a 
rule that assigns each element of the set A to a unique element of the set 
B. The word “each” specifies that there should not be ,even then single 
element in set A which has not been assigned to any of the elements of 
set B. Although it can happen that some of the elements of set B have not 
been attached to any element of set A. Set A is called as the Domain ,set 
B is called as Co-Domain and all the elements of set B, that have some 
attachment in set A, forms Range of the mapping. 
3.3. Binary Operation: Let A and B be two non empty sets then 
A×B={(a, b): a ?A and b ?B} is known as the Cartesian products of A and 
B. A binary operation is a mapping from A×A to A,  i.e. f: A×A ?A . If * is 
the binary operation then we write f(a, b)=a*b, where a, b ?A. The Binary 
operation is divided into two operations depending on the Cartesian 
products. If the Cartesian product is done between the same sets then 
the operation is known as “Internal Binary Composition” and if the 
Cartesian product is done between two distinct sets then the binary 
operation is known as “External Binary Composition”. 
  f: A×A ?A          [ Internal Binary Composition ] 
 f: A×B ?A          [ External Binary Composition ] 
3.4. Algebraic Structure: A set equipped with one or more binary 
operations is known as Algebraic Structure. An Algebraic Structure is 
denoted as : (Name of the set, First operation, Second operation, Third 
Operation, . . .) 
For example: (Z, +), (Z, +,*), (Q, +,*), (R, +,*) etc. 
 
3.5. Group : An Algebraic Structure G with only one binary operation * is 
called a Group if it satisfies the following postulates: 
 (1). G is closed under the operation *.  That is, a*b ?G, ?a, b ?G . 
 (2). The operation * is Associative. That is a*(b*c)=(a*b)*c, ?a, b, 
        c ?G 
Page 5


Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Spaces 
Paper: Linear Algebra 
Lesson Developer: Pushpendra Kumar Vashishtha and 
Dr. Arvind 
College/Department: Kamala Nehru College (D.U) / 
Hansraj College (D.U), University of Delhi 
 
 
 
 
 
 
  
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of Contents:    
 
Chapter : Vector Spaces                                                   
? Learning Outcomes 
? Introduction  
? Preliminaries 
? Vector Spaces 
? Axioms 
? Examples 
? Properties 
? Subspaces 
? Linear combination 
? Linear Span 
? Row spaces 
? Summary  
? Multiple Choice Questions 
? References 
 
 
 
1. Learning Outcomes: 
After taking a visit of this chapter the reader will be able to learn: 
? Set and Binary operations 
? Algebraic Structure 
? Vector spaces 
? Sub spaces 
? Linear Span 
? Row Spaces 
 
 
 
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
2. Introduction:  
 Mathematics is the game of numbers. What if there is no number to 
play with? O.K then we will be choosing some objects, as we are keenly 
interested in playing the game. This is reason why Mathematics has been 
divided in two parts,  One in which we do some calculations and is more 
realistic so called Applied Mathematics , other in which the things or the 
objects does not appear and we play an abstract game , is called as Pure 
Mathematics. Pure mathematics is a tree and Linear algebra is one of its 
branches. In Linear Algebra we take some objects and then spread them 
by applying some operations to make some other new objects and 
continue this process until we get a family of objects which is complete in 
itself and further cannot be expanded. The expansion of objects is done 
linearly in form of linear combination that’s why this branch is called 
Linear Algebra.  
 In this chapter we will be discussing the foundation of Linear 
Algebra so called Vector Spaces. It is hard to overstate the importance of 
the idea of a vector space, a concept which has found application in the 
areas of mathematics, engineering, physics, chemistry, biology, the social 
sciences and others.  
3. Preliminaries: 
 Before going on war we need weapons. you are here to fight with 
vectors in their homes named as “Vector Spaces” , so we first learn some 
of the basic concepts that are very essential for learning Vector Spaces. 
We will not be going into depth as it is assumed that the reader is familiar 
with these concepts. 
3.1. Set : A set is well defined collection of objects. The term well defined 
specifies that there must be some rule with the help of which we can 
unambiguously say that the particular element belongs to that set or not.  
? Sets are always denoted by the capital letters and the elements are 
denoted by the small letters. 
? The elements of a set are written in the braces {} separated by 
commas. 
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
For Example: Set of Natural Numbers, N={1,2,3,4,. . . .} 
For Example: Set of Integers Z={. . . ,-3,-2,-1,0,1,2,3, . . .} 
3.2. Mapping: A mapping, between two non empty sets A and B, is a 
rule that assigns each element of the set A to a unique element of the set 
B. The word “each” specifies that there should not be ,even then single 
element in set A which has not been assigned to any of the elements of 
set B. Although it can happen that some of the elements of set B have not 
been attached to any element of set A. Set A is called as the Domain ,set 
B is called as Co-Domain and all the elements of set B, that have some 
attachment in set A, forms Range of the mapping. 
3.3. Binary Operation: Let A and B be two non empty sets then 
A×B={(a, b): a ?A and b ?B} is known as the Cartesian products of A and 
B. A binary operation is a mapping from A×A to A,  i.e. f: A×A ?A . If * is 
the binary operation then we write f(a, b)=a*b, where a, b ?A. The Binary 
operation is divided into two operations depending on the Cartesian 
products. If the Cartesian product is done between the same sets then 
the operation is known as “Internal Binary Composition” and if the 
Cartesian product is done between two distinct sets then the binary 
operation is known as “External Binary Composition”. 
  f: A×A ?A          [ Internal Binary Composition ] 
 f: A×B ?A          [ External Binary Composition ] 
3.4. Algebraic Structure: A set equipped with one or more binary 
operations is known as Algebraic Structure. An Algebraic Structure is 
denoted as : (Name of the set, First operation, Second operation, Third 
Operation, . . .) 
For example: (Z, +), (Z, +,*), (Q, +,*), (R, +,*) etc. 
 
3.5. Group : An Algebraic Structure G with only one binary operation * is 
called a Group if it satisfies the following postulates: 
 (1). G is closed under the operation *.  That is, a*b ?G, ?a, b ?G . 
 (2). The operation * is Associative. That is a*(b*c)=(a*b)*c, ?a, b, 
        c ?G 
Vector Spaces 
Institute of Lifelong Learning, University of Delhi                                                 pg. 5 
 
 (3). Existence of Identity.; i.e. ?an element e ? G such that  
   a*e=a=e*a 
 (4). Existence of Inverse. i.e. for each element a ?G, ?an element a
-
   1
?G such that a*a
-1
=e=a
-1
*a , a
-1
 is called the inverse of a. 
If a*b=b*a ?a, b ?G then the group G is called as the Abelian Group. 
For example: (Z, +), (R, +), (C,+) with usual addition are groups. 
3.6. Field : An Algebraic Structure F with two binary operations + and * 
is called a Field if the following axioms are satisfied: 
(A). (F, +) is an Abelian group i.e. 
 (1). F is closed under the operation +.  That is a+b ?G, ?a, b ?G. 
 (2). the operation + is Associative. That is a+(b+c)=(a+b)+c, ? 
   a,b,c ?G 
 (3). Existence of Identity. i.e. ?an element 0 ? G such that  
   a+0=a=0+a 
 (4). Existence of Inverse. i.e. for each element a ?G, ?an element -
    a ?G  such  that a+(-a)=0=(-a)+a , -a is called the inverse of  
    a. 
 (5). G is commutative. i.e. a+b=b+a 
?
a, b
?
F 
(B). (F - {0},*) is an Abelian group i.e. 
 (1). F is closed under the operation *. That is a*b
?
G, 
?
a, b
?
G . 
 (2). The operation * is Associative. That is a*(b*c)=(a*b)*c,
?
a, b, 
        c 
?
G 
 (3). Existence of Identity. i.e.  ? An element e 
?
 G such that         
    a*e=a=e*a 
 (4). Existence of Inverse. i.e. for each element a 
?
G,
?
an element a
-
          1
?
G such that a*a
-1
=e=a
-1
*a , a
-1
 is called the inverse of  a.