Lecture 5 - Vector Spaces: Basis and Dimensions | Linear Algebra - Engineering Mathematics PDF Download

 Page 1


Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Vector Spaces: Basis and Dimensions 
Course Developer: Vivek N Sharma 
College / Department: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
Page 2


Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Vector Spaces: Basis and Dimensions 
Course Developer: Vivek N Sharma 
College / Department: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
Table of Contents  
1. Learning Outcomes 
2. Introduction                                                                     
3. Definition of a Vector Space   
4. Linear Combination and Span 
5. Linear Independence 
6. Basis of a Vector Space 
7. Finite Dimensional Vector Space 
8. Rank and System of Linear Equations 
9. Direct Sum of Vector Spaces 
10. Quotient of Vector Spaces 
11. Summary 
12. Glossary 
13. Further Reading 
14. Solutions/Hints for Exercises 
 
  
Page 3


Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Vector Spaces: Basis and Dimensions 
Course Developer: Vivek N Sharma 
College / Department: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
Table of Contents  
1. Learning Outcomes 
2. Introduction                                                                     
3. Definition of a Vector Space   
4. Linear Combination and Span 
5. Linear Independence 
6. Basis of a Vector Space 
7. Finite Dimensional Vector Space 
8. Rank and System of Linear Equations 
9. Direct Sum of Vector Spaces 
10. Quotient of Vector Spaces 
11. Summary 
12. Glossary 
13. Further Reading 
14. Solutions/Hints for Exercises 
 
  
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
1. Learning Outcomes 
After studying this unit, you will be able to 
 
? explain the concept of a vector space over a field. 
? understand the concept of  linear independence of vectors over a field. 
? elaborate the idea of a finite dimensional vector space. 
? state the meaning of a basis of a vector space. 
? define the dimension of a vector space. 
? define the concept of rank of a matrix. 
? analyse a system of linear equations. 
? explain the concept of direct sum of vector spaces. 
? describe the idea of a quotient of two vector spaces. 
 
2. Introduction: 
In this unit, we shall be studying one of the most important algebraic structures in 
mathematics. They are called Vector Spaces. Vector spaces are introduced in algebra. Of 
course, their applications abound. This unit gives the first introduction to these structures. 
The unit we are going to study can alternatively be referred to as „Introduction to Linear 
Algebra?. To make sense to this alternative title, it is imperative that the meaning of the 
terms 'linear? and „algebra? be clarified in the context of mathematics. The term „linear? in 
the context of algebra refers to entities which can be added in a manner „similar? to the 
addition of matrices; and which can be multiplied by numbers (scalars). Obviously, only like 
quantities can be added to each other. (For instance, one cannot add a 2 × 3 matrix to a 
4 × 4 matrix.) All such like entities when brought together constitute a Vector Space. So, 
let us commence with the idea of a vector space. 
 
 
 
 
 
 
 
 
Page 4


Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Vector Spaces: Basis and Dimensions 
Course Developer: Vivek N Sharma 
College / Department: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
Table of Contents  
1. Learning Outcomes 
2. Introduction                                                                     
3. Definition of a Vector Space   
4. Linear Combination and Span 
5. Linear Independence 
6. Basis of a Vector Space 
7. Finite Dimensional Vector Space 
8. Rank and System of Linear Equations 
9. Direct Sum of Vector Spaces 
10. Quotient of Vector Spaces 
11. Summary 
12. Glossary 
13. Further Reading 
14. Solutions/Hints for Exercises 
 
  
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
1. Learning Outcomes 
After studying this unit, you will be able to 
 
? explain the concept of a vector space over a field. 
? understand the concept of  linear independence of vectors over a field. 
? elaborate the idea of a finite dimensional vector space. 
? state the meaning of a basis of a vector space. 
? define the dimension of a vector space. 
? define the concept of rank of a matrix. 
? analyse a system of linear equations. 
? explain the concept of direct sum of vector spaces. 
? describe the idea of a quotient of two vector spaces. 
 
2. Introduction: 
In this unit, we shall be studying one of the most important algebraic structures in 
mathematics. They are called Vector Spaces. Vector spaces are introduced in algebra. Of 
course, their applications abound. This unit gives the first introduction to these structures. 
The unit we are going to study can alternatively be referred to as „Introduction to Linear 
Algebra?. To make sense to this alternative title, it is imperative that the meaning of the 
terms 'linear? and „algebra? be clarified in the context of mathematics. The term „linear? in 
the context of algebra refers to entities which can be added in a manner „similar? to the 
addition of matrices; and which can be multiplied by numbers (scalars). Obviously, only like 
quantities can be added to each other. (For instance, one cannot add a 2 × 3 matrix to a 
4 × 4 matrix.) All such like entities when brought together constitute a Vector Space. So, 
let us commence with the idea of a vector space. 
 
 
 
 
 
 
 
 
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 4 
 
3. Definition of a Vector Space: 
Before formally defining a vector space, we shall relook at R
?? for ?? = 2, 3 . We begin with 
?? = 2.The Cartesian space R
2
, which we think of as the usual ?? - ?? plane, equals the set of all 
ordered pairs of real numbers. That is, we have 
 R
2
= { ?? ,?? :?? ,?? ?? R}. 
Similarly, we have 
 R
3
= { ?? ,?? ,?? :?? ,?? ,?? ?? R}. 
Now, the question is: How, if possible, can we perform addition on the elements of R
2
?? ?? R
3
? 
The answer, as expected, emerges from co-ordinate geometry. We define addition on R
2
 
and R
3
 in a coordinate-wise manner: 
 (?? 1
,?? 1
) + ?? 2
,?? 2
 = (?? 1
+ ?? 2
,?? 1
+ ?? 2
) 
and 
 (?? 1
,?? 1
,?? 1
) + ?? 2
,?? 2
,?? 2
 = (?? 1
+ ?? 2
,?? 1
+ ?? 2
,?? 1
+ ?? 2
). 
This definition is easily adopted to R
?? : 
 (?? 1
,… … .?? ?? ) + ?? 1
,… … .?? ?? = (?? 1
+ ?? 1
,… . .?? ?? + ?? ?? ) 
and, hence, on F
?? where F = R or C. We may note that when F = C, we have  
 F
?? = C
?? = { ?? 1
,… . ,?? ?? : ?? ?? ?? C ? ?? = 1 1 ?? } 
The notation ?? = 1 1 ?? stands for: “the index ?? starts from one, raised by one, upto ?? ”, that 
is, ?? = 1,2,3,… … ,?? . Writing out elements of F
?? explicity in a co-ordinate-wise manner may not 
be illuminating every time. So, to denote the elements of F
?? , we compress the ?? -co-
ordinates to a single symbol. We simply write: 
?? = ?? 1
,?? 2
… . . ,?? ?? . 
Hence, addition on F
?? corresponds to 
 ?? + ?? = ?? 1
,… .?? ?? +  ?? 1
,… .?? ?? = (?? 1
+ ?? 1
,… .?? ?? + ?? ?? ). 
The other algebraic operation, namely, scalar multiplication, is defined as follows: 
 ?? .?? = ?? ?? 1
,… .?? ?? = ?? ?? 1
,… .?? ?? ?? . 
 
 
 
 
 
 
 
 
 
Page 5


Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: Vector Spaces: Basis and Dimensions 
Course Developer: Vivek N Sharma 
College / Department: Assistant Professor, Department of 
Mathematics, S.G.T.B. Khalsa College, University of Delhi 
 
 
 
 
 
  
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
Table of Contents  
1. Learning Outcomes 
2. Introduction                                                                     
3. Definition of a Vector Space   
4. Linear Combination and Span 
5. Linear Independence 
6. Basis of a Vector Space 
7. Finite Dimensional Vector Space 
8. Rank and System of Linear Equations 
9. Direct Sum of Vector Spaces 
10. Quotient of Vector Spaces 
11. Summary 
12. Glossary 
13. Further Reading 
14. Solutions/Hints for Exercises 
 
  
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
1. Learning Outcomes 
After studying this unit, you will be able to 
 
? explain the concept of a vector space over a field. 
? understand the concept of  linear independence of vectors over a field. 
? elaborate the idea of a finite dimensional vector space. 
? state the meaning of a basis of a vector space. 
? define the dimension of a vector space. 
? define the concept of rank of a matrix. 
? analyse a system of linear equations. 
? explain the concept of direct sum of vector spaces. 
? describe the idea of a quotient of two vector spaces. 
 
2. Introduction: 
In this unit, we shall be studying one of the most important algebraic structures in 
mathematics. They are called Vector Spaces. Vector spaces are introduced in algebra. Of 
course, their applications abound. This unit gives the first introduction to these structures. 
The unit we are going to study can alternatively be referred to as „Introduction to Linear 
Algebra?. To make sense to this alternative title, it is imperative that the meaning of the 
terms 'linear? and „algebra? be clarified in the context of mathematics. The term „linear? in 
the context of algebra refers to entities which can be added in a manner „similar? to the 
addition of matrices; and which can be multiplied by numbers (scalars). Obviously, only like 
quantities can be added to each other. (For instance, one cannot add a 2 × 3 matrix to a 
4 × 4 matrix.) All such like entities when brought together constitute a Vector Space. So, 
let us commence with the idea of a vector space. 
 
 
 
 
 
 
 
 
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 4 
 
3. Definition of a Vector Space: 
Before formally defining a vector space, we shall relook at R
?? for ?? = 2, 3 . We begin with 
?? = 2.The Cartesian space R
2
, which we think of as the usual ?? - ?? plane, equals the set of all 
ordered pairs of real numbers. That is, we have 
 R
2
= { ?? ,?? :?? ,?? ?? R}. 
Similarly, we have 
 R
3
= { ?? ,?? ,?? :?? ,?? ,?? ?? R}. 
Now, the question is: How, if possible, can we perform addition on the elements of R
2
?? ?? R
3
? 
The answer, as expected, emerges from co-ordinate geometry. We define addition on R
2
 
and R
3
 in a coordinate-wise manner: 
 (?? 1
,?? 1
) + ?? 2
,?? 2
 = (?? 1
+ ?? 2
,?? 1
+ ?? 2
) 
and 
 (?? 1
,?? 1
,?? 1
) + ?? 2
,?? 2
,?? 2
 = (?? 1
+ ?? 2
,?? 1
+ ?? 2
,?? 1
+ ?? 2
). 
This definition is easily adopted to R
?? : 
 (?? 1
,… … .?? ?? ) + ?? 1
,… … .?? ?? = (?? 1
+ ?? 1
,… . .?? ?? + ?? ?? ) 
and, hence, on F
?? where F = R or C. We may note that when F = C, we have  
 F
?? = C
?? = { ?? 1
,… . ,?? ?? : ?? ?? ?? C ? ?? = 1 1 ?? } 
The notation ?? = 1 1 ?? stands for: “the index ?? starts from one, raised by one, upto ?? ”, that 
is, ?? = 1,2,3,… … ,?? . Writing out elements of F
?? explicity in a co-ordinate-wise manner may not 
be illuminating every time. So, to denote the elements of F
?? , we compress the ?? -co-
ordinates to a single symbol. We simply write: 
?? = ?? 1
,?? 2
… . . ,?? ?? . 
Hence, addition on F
?? corresponds to 
 ?? + ?? = ?? 1
,… .?? ?? +  ?? 1
,… .?? ?? = (?? 1
+ ?? 1
,… .?? ?? + ?? ?? ). 
The other algebraic operation, namely, scalar multiplication, is defined as follows: 
 ?? .?? = ?? ?? 1
,… .?? ?? = ?? ?? 1
,… .?? ?? ?? . 
 
 
 
 
 
 
 
 
 
Vector Spaces: Basis and Dimensions 
Institute of Lifelong Learning, University of Delhi                                                      pg. 5 
 
The general concept of a vector space follows this model of Euclidean space R
?? . So, we may 
now give the formal definition of a vector space. 
 
Definition of Vector Space: We first remark that by an addition on ?? , we mean a function 
on ?? × ?? that assigns an element ?? + ?? in ?? to every pair (?? ,?? ) in ?? × ?? . Further, by a scalar 
multiplication on ?? , we mean a function on F × ?? that assigns an element ?? ?? in ?? to each 
pair (?? ,?? ) in F × ?? , where F is a field. The formal definition of a vector space is as under: 
A vector space is a set ?? along with two operations: an addition on ?? and a scalar 
multiplication on ?? such that the following eight properties hold for all ?? ,?? ,?? in ?? & for all 
?? ,?? in F:  
1. ?? + ?? = ?? + ?? (Commutativity); 
2. (?? + ?? ) + ?? = ?? + (?? + ?? ) (Associativity); 
3. There exists an element 0 in ?? such that ?? + 0 = ?? (Additive Identity); 
4. for every ?? in ?? , there exists a ?? in ?? such that ?? + ?? = 0 (Additive Inverse). We 
may denote ?? by –?? , that is ?? = -?? . 
5. 1.?? = ?? ; 
6. (?? ?? )?? = ?? (?? ?? ) (Associativity); 
7. ?? (?? + ?? ) = ?? ?? + ?? ?? (Distributivity);  
8. (?? + ?? )?? = ?? ?? + ?? ?? (Distributivity). 
Hence, in scalar multiplication, scalars come from the field F. Thus, this scalar multiplication 
depends upon field F. Therefore, with reference to a vector space ?? , we say that “?? is a 
vector space over a field F” (instead of simply saying that ?? is a vector space). There are 
various notations available to express the same ?? F
,   ?? (?? ) and so on. For our purpose, we 
shall work with F = R or C.  
 
More often than not, the choice of F is obvious from the context. It goes without explicit 
mention. Let us now turn to some of the famous examples of vector spaces. 
 
 
 
  Value Addition : Remark 
When F = R, the vector space ?? over R is called real vector space and, when 
F = C, the vector space ?? over C is called a complex vector space.