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Algebra I: A Basic Course in Abstract Algebra

Book Description

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.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. About the Authors
  5. Dedication
  6. Preface
  7. Unit - 1
    1. 1. Sets and Relations
      1. 1.1 Sets
      2. 1.2 Exercise
      3. 1.3 Algebra of Sets
      4. 1.4 Exercise
      5. 1.5 Binary Relation
      6. 1.6 Exercise
      7. 1.7 Supplementary Exercises
      8. 1.8 Answers to Exercises
    2. 2. Binary Operations
      1. 2.1 Definition and Examples
      2. 2.2 Exercise
      3. 2.3 Introduction to Groups
      4. 2.4 Symmetries
      5. 2.5 Exercise
      6. 2.6 Solved Problems
      7. 2.7 Supplementary Exercises
      8. 2.8 Answers to Exercises
    3. 3. Functions
      1. 3.1 Definition and Representation
      2. 3.2 Images and Inverse Images
      3. 3.3 Types of Functions
      4. 3.4 Real Valued Functions
      5. 3.5 Some Functions on the Set of Real Numbers
      6. 3.6 Exercise
      7. 3.7 Inverse of a Function
      8. 3.8 Composition of Functions
      9. 3.9 Solved Problems
      10. 3.10 Exercise
      11. 3.11 Cardinality of a Set
      12. 3.12 Countable Sets
      13. 3.13 Exercise
      14. 3.14 Solved Problems
      15. 3.15 Supplementary Exercise
      16. 3.16 Answers to Exercises
    4. 4. Number System
      1. 4.1 Number Systems
      2. 4.2 Division Algorithm
      3. 4.3 Exercise
      4. 4.4 Greatest Common Divisor
      5. 4.5 Least Common Multiple
      6. 4.6 Exercise
      7. 4.7 Congruence Relation
      8. 4.8 Exercise
      9. 4.9 Supplementary Problems
      10. 4.10 Answers to Exercises
  8. Unit - 2
    1. 5. Group: Definition and Examples
      1. 5.1 Definition of Group
      2. 5.2 Exercise
      3. 5.3 Groups of Numbers
      4. 5.4 Exercise
      5. 5.5 Groups of Residues
      6. 5.6 Exercise
      7. 5.7 Groups of Matrices
      8. 5.8 Exercise
      9. 5.9 Groups of Functions
      10. 5.10 Exercise
      11. 5.11 Group of Subsets of a Set
      12. 5.12 Exercise
      13. 5.13 Groups of Symmetries
      14. 5.14 Supplementary Exercise
      15. 5.15 Answers to Exercises
    2. 6. Group: Properties and Characterization
      1. 6.1 Properties of Groups
      2. 6.2 Solved Problems
      3. 6.3 Exercise
      4. 6.4 Characterization of Groups
      5. 6.5 Solved Problems
      6. 6.6 Exercise
      7. 6.7 Supplementary Exercises
      8. 6.8 Answers to Exercises
    3. 7. Subgroups
      1. 7.1 Criteria for Subgroups
      2. 7.2 Solved Problems
      3. 7.3 Exercise
      4. 7.4 Centralizers, Normalizers and Centre
      5. 7.5 Exercise
      6. 7.6 Order of an Element
      7. 7.7 Solved Problems
      8. 7.8 Exercise
      9. 7.9 Cyclic Subgroups
      10. 7.10 Solved Problems
      11. 7.11 Exercise
      12. 7.12 Lattice of Subgroups
      13. 7.13 Exercise
      14. 7.14 Supplementary Exercises
      15. 7.15 Answers to Exercises
    4. 8. Cyclic Groups
      1. 8.1 Definition and Examples
      2. 8.2 Description of Cyclic Groups
      3. 8.3 Exercise
      4. 8.4 Generators of a Cyclic Group
      5. 8.5 Exercise
      6. 8.6 Subgroups of Cyclic Groups
      7. 8.7 Subgroups of Infinite Cyclic Groups
      8. 8.8 Subgroups of Finite Cyclic Groups
      9. 8.9 Number of Generators
      10. 8.10 Exercise
      11. 8.11 Solved Problems
      12. 8.12 Supplementary Exercise
      13. 8.13 Answers to Exercises
  9. Unit - 3
    1. 9. Rings
      1. 9.1 Ring
      2. 9.2 Examples of Ring
      3. 9.3 Constructing New Rings
      4. 9.4 Special Elements of a Ring
      5. 9.5 Solved Problems
      6. 9.6 Exercise
      7. 9.7 Subrings
      8. 9.8 Exercise
      9. 9.9 Integral Domains and Fields
      10. 9.10 Examples
      11. 9.11 Exercise
      12. 9.12 Solved Problems
      13. 9.13 Supplementary Exercises
      14. 9.14 Answers to Exercise
  10. Unit - 4
    1. 10 System of Linear Equations
      1. 10.1 Matrix Notation
      2. 10.2 Solving a Linear System
      3. 10.3 Elementary Row Operations (ERO)
      4. 10.4 Solved Problems
      5. 10.5 Exercise
      6. 10.6 Row Reduction and Echelon Forms
      7. 10.7 Exercise
      8. 10.8 Vector Equations
      9. 10.9 Vectors in R2
      10. 10.10 Geometric Descriptions of R2
      11. 10.11 Vectors in Rn
      12. 10.12 Exercise
      13. 10.13 Solutions of Linear Systems
      14. 10.14 Parametric Description of Solution Sets
      15. 10.15 Existence and Uniqueness of Solutions
      16. 10.16 Homogenous System
      17. 10.17 Exercise
      18. 10.18 Solution Sets of Linear Systems
      19. 10.19 Exercise
      20. 10.20 Answers to Exercises
    2. 11. Matrices
      1. 11.1 Matrix of Numbers
      2. 11.2 Operations on Matrices
      3. 11.3 Partitioning of Matrices
      4. 11.4 Special Types of Matrices
      5. 11.5 Exercise
      6. 11.6 Inverse of a Matrix
      7. 11.7 Adjoint of a Matrix
      8. 11.8 Negative Integral Powers of a Non-singular Matrix
      9. 11.9 Inverse of Partitioned Matrices
      10. 11.10 Solved Problems
      11. 11.11 Exercise
      12. 11.12 Orthogonal and Unitary Matrices
      13. 11.13 Length Preserving Mapping
      14. 11.14 Exercise
      15. 11.15 Eigenvalues and Eigenvectors
      16. 11.16 Cayley Hamilton Theorem and Its Applications
      17. 11.17 Solved Problems
      18. 11.18 Exercise
      19. 11.19 Supplementary Exercises
      20. 11.20 Answers to Exercises
    3. 12. Matrices and Linear Transformations
      1. 12.1 Introduction to Linear Transformations
      2. 12.2 Exercise
      3. 12.3 Matrix Transformations
      4. 12.4 Surjective and Injective Matrix Transformations
      5. 12.5 Exercise
      6. 12.6 Linear Transformation
      7. 12.7 Exercise
      8. 12.8 The Matrix of a Linear Transformation
      9. 12.9 Exercises
      10. 12.10 Geometric Transformations of R2 and R3
      11. 12.11 Exercises
      12. 12.12 Supplementary Problems
      13. 12.13 Supplementary Exercise
      14. 12.14 Answers to Exercises
  11. Unit - 5
    1. 13. Vector Space
      1. 13.1 Definition and Examples
      2. 13.2 Exercise
      3. 13.3 Subspaces
      4. 13.4 Exercise
      5. 13.5 Linear Span of a Subset
      6. 13.6 Column Space
      7. 13.7 Exercise
      8. 13.8 Solved Problems
      9. 13.9 Exercise
      10. 13.10 Answers to Exercises
    2. 14. Basis and Dimension
      1. 14.1 Linearly Dependent Sets
      2. 14.2 Solved Problems
      3. 14.3 Exercise
      4. 14.4 Basis of Vector Space
      5. 14.5 Coordinates Relative to an Ordered Basis
      6. 14.6 Exercise
      7. 14.7 Dimension
      8. 14.8 Rank of a Matrix
      9. 14.9 Exercise
      10. 14.10 Solved Problems
      11. 14.11 Supplementary Exercises
      12. 14.12 Answers to Exercises
    3. 15. Linear Transformation
      1. 15.1 Definitions and Examples
      2. 15.2 Exercise
      3. 15.3 Range and Kernel
      4. 15.4 Exercise
      5. 15.5 Answers to Exercises
    4. 16. Change of Basis
      1. 16.1 Coordinate Mapping
      2. 16.2 Change of Basis
      3. 16.3 Procedure to Compute Transition Matrix PB B from Basis B1 to Basis B2
      4. 16.4 Exercise
      5. 16.5 Matrix of a Linear Transformation
      6. 16.6 Working Rule to Obtain [T]B1B2
      7. 16.7 Exercise
      8. 16.8 Supplementary Exercises
      9. 16.9 Answers to Exercises
    5. 17. Eigenvectors and Eigenvalues
      1. 17.1 Eigenvectors and Eigenspace
      2. 17.2 Solved Problems
      3. 17.3 Exercise
      4. 17.4 Characteristic Equation
      5. 17.5 Exercise
      6. 17.6 Diagonalization
      7. 17.7 Exercise
      8. 17.8 Supplementary Exercises
      9. 17.9 Answers to Exercises
    6. 18. Markov Process
      1. 18.1 Exercise
      2. 18.2 Answers to Exercises
  12. Notes
  13. Copyright