Algebra I: a Basic Course in Abstract Algebra

✍ Scribed by Shah, Sudesh Kumari; Shankar, Asha Gauri; Sharma, Rajendra Kumar

• Publisher: Pearson Education India
• Year: 2011
• Tongue: English
• Leaves: 780
• Category: Library

✦ Synopsis

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 semester. In views of this, we are bringing out three books ranging from introductory to advance level course in Algebra.Algebra I is the first book of the series and covers the topic required for a basic course.

✦ Table of Contents

Cover......Page 1
Contents......Page 6
Preface......Page 14
About the Authors......Page 16
Unit - 1......Page 18
1.1 Sets......Page 19
1.2 Exercise......Page 23
1.3 Algebra of Sets......Page 24
1.4 Exercise......Page 35
1.5 Binary Relation......Page 37
Graph of a Relation......Page 38
Properties of Binary Relation on a Set......Page 39
Equivalence Relation......Page 44
Graph of an Equivalence Relation......Page 47
1.6 Exercise......Page 50
1.7 Supplementary Exercises......Page 54
1.8 Answers to Exercises......Page 57
2.1 Definition and Examples......Page 66
The multiplication table (Cayley table)......Page 67
Properties of binary operations......Page 68
Operation with Identity Element......Page 69
2.2 Exercise......Page 71
2.3 Introduction to Groups......Page 72
Symmetries of Non-square Rectangle......Page 75
Symmetries of an Equilateral Triangle......Page 76
Dihedral group......Page 78
2.5 Exercise......Page 80
2.6 Solved Problems......Page 82
2.7 Supplementary Exercises......Page 85
2.8 Answers to Exercises......Page 89
3.1 Definition and Representation......Page 94
Arrow Diagram for Function......Page 95
Representation of a Function......Page 96
3.2 Images and Inverse Images......Page 97
Inverse image of a set......Page 98
3.3 Types of Functions......Page 100
3.4 Real Valued Functions......Page 103
3.5 Some Functions on the Set of Real Numbers......Page 105
3.6 Exercise......Page 112
3.7 Inverse of a Function......Page 114
3.8 Composition of Functions......Page 117
3.9 Solved Problems......Page 122
3.10 Exercise......Page 126
3.11 Cardinality of a Set......Page 127
3.12 Countable Sets......Page 129
3.13 Exercise......Page 138
3.14 Solved Problems......Page 139
3.15 Supplementary Exercise......Page 142
3.16 Answers to Exercises......Page 144
Algebraic Properties of Natural Numbers......Page 151
Order Properties of Natural Numbers......Page 152
Order Properties of Integers......Page 153
Divisibility......Page 157
4.2 Division Algorithm......Page 158
4.3 Exercise......Page 165
4.4 Greatest Common Divisor......Page 167
Euclidean Algorithm......Page 171
Working Rule......Page 173
4.5 Least Common Multiple......Page 176
4.6 Exercise......Page 179
4.7 Congruence Relation......Page 181
4.8 Exercise......Page 191
4.9 Supplementary Problems......Page 194
4.10 Answers to Exercises......Page 195
Unit - 2......Page 200
5.1 Definition of Group......Page 201
5.3 Groups of Numbers......Page 204
5.4 Exercise......Page 206
5.5 Groups of Residues......Page 207
5.6 Exercise......Page 210
5.7 Groups of Matrices......Page 211
5.9 Groups of Functions......Page 214
5.11 Group of Subsets of a Set......Page 216
5.13 Groups of Symmetries......Page 217
5.14 Supplementary Exercise......Page 221
5.15 Answers to Exercises......Page 224
6.1 Properties of Groups......Page 228
6.2 Solved Problems......Page 232
6.3 Exercise......Page 236
6.4 Characterization of Groups......Page 237
6.5 Solved Problems......Page 240
6.6 Exercise......Page 244
6.7 Supplementary Exercises......Page 245
6.8 Answers to Exercises......Page 246
7.1 Criteria for Subgroups......Page 248
7.2 Solved Problems......Page 252
7.3 Exercise......Page 254
Centralizer of an Element......Page 255
Centralizer of a Subset......Page 257
Centre of a Group......Page 258
Normalizer of a subset......Page 260
7.5 Exercise......Page 261
7.6 Order of an Element......Page 262
7.7 Solved Problems......Page 266
7.8 Exercise......Page 268
7.9 Cyclic Subgroups......Page 270
7.10 Solved Problems......Page 272
7.12 Lattice of Subgroups......Page 274
7.13 Exercise......Page 279
7.14 Supplementary Exercises......Page 280
7.15 Answers to Exercises......Page 282
8.1 Definition and Examples......Page 288
8.2 Description of Cyclic Groups......Page 290
8.4 Generators of a Cyclic Group......Page 293
8.5 Exercise......Page 295
8.7 Subgroups of Infinite Cyclic Groups......Page 297
8.8 Subgroups of Finite Cyclic Groups......Page 298
8.9 Number of Generators......Page 300
8.10 Exercise......Page 303
8.11 Solved Problems......Page 304
8.12 Supplementary Exercise......Page 307
8.13 Answers to Exercises......Page 309
Unit - 3......Page 314
9.1 Ring......Page 315
Rings of Residues......Page 316
Ring of polynomials......Page 317
Elementary Properties of Ring......Page 318
9.3 Constructing New Rings......Page 320
9.4 Special Elements of a Ring......Page 321
9.5 Solved Problems......Page 322
Solution:......Page 323
9.6 Exercise......Page 326
Criterion for a subset to be a subring......Page 329
Examples from Matrices......Page 337
Example from Quaternions......Page 338
9.8 Exercise......Page 333
9.9 Integral Domains and Fields......Page 335
9.10 Examples......Page 336
9.11 Exercise......Page 343
9.12 Solved Problems......Page 344
9.13 Supplementary Exercises......Page 345
9.14 Answers to Exercise......Page 349
Unit - 4......Page 354
Chapter 10: System of Linear Equations......Page 355
Geometrical Interpretation......Page 356
10.1 Matrix Notation......Page 357
10.2 Solving a Linear System......Page 358
10.3 Elementary Row Operations (ERO)......Page 359
10.4 Solved Problems......Page 361
10.5 Exercise......Page 370
10.6 Row Reduction and Echelon Forms......Page 374
10.7 Exercise......Page 388
10.8 Vector Equations......Page 390
10.10 Geometric Descriptions of R2......Page 391
Algebraic Properties of Rn......Page 394
Lines in Rn......Page 395
Linear Combination of Vectors......Page 396
10.12 Exercise......Page 405
10.13 Solutions of Linear Systems......Page 410
10.14 Parametric Description of Solution Sets......Page 414
10.15 Existence and Uniqueness of Solutions......Page 416
10.16 Homogenous System......Page 420
10.17 Exercise......Page 433
10.18 Solution Sets of Linear Systems......Page 439
10.20 Answers to Exercises......Page 447
11.1 Matrix of Numbers......Page 458
On the Basis of Elements......Page 459
11.2 Operations on Matrices......Page 460
11.3 Partitioning of Matrices......Page 468
11.3.1 Multiplication of Partitioned Matrices......Page 470
Symmetric and Skew Symmetric Matrices......Page 472
Hermitian and Skew Hermitian Matrices......Page 474
11.5 Exercise......Page 477
11.6 Inverse of a Matrix......Page 480
11.7 Adjoint of a Matrix......Page 481
11.8 Negative Integral Powers of a Non-singular Matrix......Page 483
11.9 Inverse of Partitioned Matrices......Page 484
11.10 Solved Problems......Page 487
11.11 Exercise......Page 490
11.12 Orthogonal and Unitary Matrices......Page 493
11.13 Length Preserving Mapping......Page 495
11.14 Exercise......Page 498
Determination of eigenvalues and eigenvectors......Page 500
11.16 Cayley Hamilton Theorem and its Applications......Page 507
11.17 Solved Problems......Page 510
11.18 Exercise......Page 517
11.19 Supplementary Exercises......Page 518
11.20 Answers to Exercises......Page 521
12.1 Introduction to Linear Transformations......Page 526
12.3 Matrix Transformations......Page 530
12.4 Surjective and Injective Matrix Transformations......Page 531
12.5 Exercise......Page 538
12.6 Linear Transformation......Page 541
How to prove non-linearity?......Page 542
Geometrical Properties of Linear Transformation......Page 544
12.7 Exercise......Page 545
12.8 The Matrix of a Linear Transformation......Page 547
12.9 Exercises......Page 549
Scaling......Page 551
Shear Transformation......Page 556
Matrices of Geometric Linear Transformation in R2......Page 561
Geometrical Interpretation of Some Transformation......Page 566
12.11 Exercises......Page 569
12.12 Supplementary Problems......Page 570
12.13 Supplementary Exercise......Page 572
12.14 Answers to Exercises......Page 576
Unit - 5......Page 580
13.1 Definition and Examples......Page 581
Elementary Properties......Page 582
Notation......Page 583
13.2 Exercise......Page 590
13.3 Subspaces......Page 591
13.4 Exercise......Page 598
13.5 Linear Span of a Subset......Page 600
13.6 Column Space......Page 603
13.7 Exercise......Page 608
13.8 Solved Problems......Page 610
13.9 Exercise......Page 612
13.10 Answers to Exercises......Page 616
14.1 Linearly Dependent Sets......Page 618
14.2 Solved Problems......Page 624
14.3 Exercise......Page 630
14.4 Basis of Vector Space......Page 632
14.5 Coordinates Relative to an Ordered Basis......Page 634
14.6 Exercise......Page 642
14.7 Dimension......Page 644
14.8 Rank of a Matrix......Page 651
14.9 Exercise......Page 657
14.10 Solved Problems......Page 661
14.11 Supplementary Exercises......Page 662
14.12 Answers to Exercises......Page 666
15.1 Definitions and Examples......Page 670
15.2 Exercise......Page 679
15.3 Range and Kernel......Page 681
15.4 Exercise......Page 690
15.5 Answers to Exercises......Page 693
16.1 Coordinate Mapping......Page 695
16.2 Change of Basis......Page 696
16.3 Procedure to Compute Transition Matrix PB B from Basis B1 to Basis B2......Page 700
16.4 Exercise......Page 704
16.5 Matrix of a Linear Transformation......Page 708
16.6 Working Rule to Obtain [T]B1B2......Page 710
16.7 Exercise......Page 719
16.8 Supplementary Exercises......Page 722
16.9 Answers to Exercises......Page 728
17.1 Eigenvectors and Eigenspace......Page 736
17.2 Solved Problems......Page 740
17.3 Exercise......Page 741
17.4 Characteristic Equation......Page 743
17.5 Exercise......Page 752
17.6 Diagonalization......Page 754
17.7 Exercise......Page 759
17.8 Supplementary Exercises......Page 760
17.9 Answers to Exercises......Page 762
Chapter 18: Markov Process......Page 765
18.1 Exercise......Page 773
18.2 Answers to Exercises......Page 775
Index......Page 778