Algebra I : a basic course in abstract algebra

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.

Detalles Bibliográficos
Otros Autores:	Sharma, Rajendra K Author (author), Shah, Sudesh Kumari Contributor (contributor), Shankar, Asha Gauri Contributor
Formato:	Libro electrónico
Idioma:	Inglés
Publicado:	[Place of publication not identified] Pearson Education India 2011
Edición:	1st edition
Colección:	Always learning.
Materias:	Mathematics Physical Sciences & Mathematics Algebra
Ver en Biblioteca Universitat Ramon Llull:	https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009628010306719

Tabla de Contenidos:

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.

Algebra I : a basic course in abstract algebra

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