Algebra I : a basic course in abstract algebra

Format:

Book

Author/Creator:

Sharma, Rajendra K., Author.

Contributor:

Shah, Sudesh Kumari, Contributor.

Shankar, Asha Gauri, Contributor.

Series:

Always learning.

Always Learning

Language:

English

Physical Description:

1 online resource (1 v.) : ill.

Edition:

1st edition

Other Title:

Algebra I

Place of Publication:

[Place of publication not identified] Pearson Education India 2011

Language Note:

English

System Details:

text file

Summary:

Algebra is a compulsory paper offered to the undergraduate students of Mathematics. The majority of universities offer the subject as a two /three year paper or in two/three semesters. Algebra I: A Basic Course in Abstract Algebra covers the topic required for a basic course.

Contents:

Cover

Contents

Preface

About the Authors

Unit - 1

Chapter 1: Sets and Relations

1.1 Sets

1.2 Exercise

1.3 Algebra of Sets

1.4 Exercise

1.5 Binary Relation

Graph of a Relation

Properties of Binary Relation on a Set

Equivalence Relation

Graph of an Equivalence Relation

1.6 Exercise

1.7 Supplementary Exercises

1.8 Answers to Exercises

Chapter 2: Binary Operations

2.1 Definition and Examples

The multiplication table (Cayley table)

Properties of binary operations

Operation with Identity Element

2.2 Exercise

2.3 Introduction to Groups

2.4 Symmetries

Symmetries of Non-square Rectangle

Symmetries of an Equilateral Triangle

Dihedral group

2.5 Exercise

2.6 Solved Problems

2.7 Supplementary Exercises

2.8 Answers to Exercises

Chapter 3: Functions

3.1 Definition and Representation

Arrow Diagram for Function

Representation of a Function

3.2 Images and Inverse Images

Inverse Images

Inverse image of a set

3.3 Types of Functions

3.4 Real Valued Functions

3.5 Some Functions on the Set of Real Numbers

3.6 Exercise

3.7 Inverse of a Function

3.8 Composition of Functions

3.9 Solved Problems

3.10 Exercise

3.11 Cardinality of a Set

3.12 Countable Sets

3.13 Exercise

3.14 Solved Problems

3.15 Supplementary Exercise

3.16 Answers to Exercises

Chapter 4: Number System

4.1 Number Systems

Algebraic Properties of Natural Numbers

Order Properties of Natural Numbers

Algebraic Properties of Integers

Order Properties of Integers

Divisibility

4.2 Division Algorithm

4.3 Exercise

4.4 Greatest Common Divisor

Euclidean Algorithm

Working Rule

4.5 Least Common Multiple

4.6 Exercise

4.7 Congruence Relation

4.8 Exercise

4.9 Supplementary Problems

4.10 Answers to Exercises.

Unit - 2

Chapter 5: Group Definition and Examples

5.1 Definition of Group

5.2 Exercise

5.3 Groups of Numbers

5.4 Exercise

5.5 Groups of Residues

5.6 Exercise

5.7 Groups of Matrices

5.8 Exercise

5.9 Groups of Functions

5.10 Exercise

5.11 Group of Subsets of a Set

5.12 Exercise

5.13 Groups of Symmetries

5.14 Supplementary Exercise

5.15 Answers to Exercises

Chapter 6: Group Properties and Characterization

6.1 Properties of Groups

6.2 Solved Problems

6.3 Exercise

6.4 Characterization of Groups

6.5 Solved Problems

6.6 Exercise

6.7 Supplementary Exercises

6.8 Answers to Exercises

Chapter 7: Subgroups

7.1 Criteria for Subgroups

7.2 Solved Problems

7.3 Exercise

7.4 Centralizers, Normalizers and Centre

Centralizer of an Element

Centralizer of a Subset

Centre of a Group

Normalizer of a subset

7.5 Exercise

7.6 Order of an Element

7.7 Solved Problems

7.8 Exercise

7.9 Cyclic Subgroups

7.10 Solved Problems

7.11 Exercise

7.12 Lattice of Subgroups

7.13 Exercise

7.14 Supplementary Exercises

7.15 Answers to Exercises

Chapter 8: Cyclic Groups

8.1 Definition and Examples

8.2 Description of Cyclic Groups

8.3 Exercise

8.4 Generators of a Cyclic Group

8.5 Exercise

8.6 Subgroups of Cyclic Groups

8.7 Subgroups of Infinite Cyclic Groups

8.8 Subgroups of Finite Cyclic Groups

8.9 Number of Generators

8.10 Exercise

8.11 Solved Problems

8.12 Supplementary Exercise

8.13 Answers to Exercises

Unit - 3

Chapter 9: Rings

9.1 Ring

9.2 Examples of Ring

Rings of Numbers

Rings of Residues

Rings of Matrices

Ring of polynomials

Ring of Functions

Elementary Properties of Ring

9.3 Constructing New Rings

9.4 Special Elements of a Ring

9.5 Solved Problems

Solution:

9.6 Exercise.

9.7 Subrings

Criterion for a subset to be a subring

Examples from Matrices

Example from Quaternions

9.8 Exercise

9.9 Integral Domains and Fields

9.10 Examples

9.11 Exercise

9.12 Solved Problems

9.13 Supplementary Exercises

9.14 Answers to Exercise

Unit - 4

Chapter 10: System of Linear Equations

Geometrical Interpretation

10.1 Matrix Notation

10.2 Solving a Linear System

10.3 Elementary Row Operations (ERO)

10.4 Solved Problems

10.5 Exercise

10.6 Row Reduction and Echelon Forms

10.7 Exercise

10.8 Vector Equations

10.9 Vectors in R2

10.10 Geometric Descriptions of R2

10.11 Vectors in Rn

Algebraic Properties of Rn

Points in Rn

Lines in Rn

Planes in Rn

Linear Combination of Vectors

10.12 Exercise

10.13 Solutions of Linear Systems

10.14 Parametric Description of Solution Sets

10.15 Existence and Uniqueness of Solutions

10.16 Homogenous System

10.17 Exercise

10.18 Solution Sets of Linear Systems

10.19 Exercise

10.20 Answers to Exercises

Chapter 11: Matrices

11.1 Matrix of Numbers

Types of matrices

On the basis of size

On the Basis of Elements

11.2 Operations on Matrices

11.3 Partitioning of Matrices

11.3.1 Multiplication of Partitioned Matrices

11.4 Special Types of Matrices

Symmetric and Skew Symmetric Matrices

Hermitian and Skew Hermitian Matrices

11.5 Exercise

11.6 Inverse of a Matrix

11.7 Adjoint of a Matrix

11.8 Negative Integral Powers of a Non-singular Matrix

11.9 Inverse of Partitioned Matrices

11.10 Solved Problems

11.11 Exercise

11.12 Orthogonal and Unitary Matrices

11.13 Length Preserving Mapping

11.14 Exercise

11.15 Eigenvalues and Eigenvectors

Determination of eigenvalues and eigenvectors

11.16 Cayley Hamilton Theorem and its Applications.

11.17 Solved Problems

11.18 Exercise

11.19 Supplementary Exercises

11.20 Answers to Exercises

Chapter 12: Matrices and Linear Transformations

12.1 Introduction to Linear Transformations

12.2 Exercise

12.3 Matrix Transformations

12.4 Surjective and Injective Matrix Transformations

12.5 Exercise

12.6 Linear Transformation

How to prove non-linearity?

Geometrical Properties of Linear Transformation

12.7 Exercise

12.8 The Matrix of a Linear Transformation

12.9 Exercises

12.10 Geometric Transformations of R2 and R3

Scaling

Shear Transformation

Matrices of Geometric Linear Transformation in R2

Geometrical Interpretation of Some Transformation

12.11 Exercises

12.12 Supplementary Problems

12.13 Supplementary Exercise

12.14 Answers to Exercises

Unit - 5

Chapter 13: Vector Space

13.1 Definition and Examples

Elementary Properties

Notation

13.2 Exercise

13.3 Subspaces

13.4 Exercise

13.5 Linear Span of a Subset

13.6 Column Space

13.7 Exercise

13.8 Solved Problems

13.9 Exercise

13.10 Answers to Exercises

Chapter 14: Basis and Dimension

14.1 Linearly Dependent Sets

14.2 Solved Problems

14.3 Exercise

14.4 Basis of Vector Space

14.5 Coordinates Relative to an Ordered Basis

14.6 Exercise

14.7 Dimension

14.8 Rank of a Matrix

14.9 Exercise

14.10 Solved Problems

14.11 Supplementary Exercises

14.12 Answers to Exercises

Chapter 15: Linear Transformation

15.1 Definitions and Examples

15.2 Exercise

15.3 Range and Kernel

15.4 Exercise

15.5 Answers to Exercises

Chapter 16: Change of Basis

16.1 Coordinate Mapping

16.2 Change of Basis

16.3 Procedure to Compute Transition Matrix PB B from Basis B1 to Basis B2

16.4 Exercise

16.5 Matrix of a Linear Transformation.

16.6 Working Rule to Obtain [T]B1B2

16.7 Exercise

16.8 Supplementary Exercises

16.9 Answers to Exercises

Chapter 17: Eigenvectors and Eigenvalues

17.1 Eigenvectors and Eigenspace

17.2 Solved Problems

17.3 Exercise

17.4 Characteristic Equation

17.5 Exercise

17.6 Diagonalization

17.7 Exercise

17.8 Supplementary Exercises

17.9 Answers to Exercises

Chapter 18: Markov Process

18.1 Exercise

18.2 Answers to Exercises

Index.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references.

Description based on publisher supplied metadata and other sources.

ISBN:

9789332515697

9332515697

9788131797624

8131797627

OCLC:

1024269108