Algebra and geometry

Alan F. Beardon9780511113277, 9780521813624, 052181362X, 0521890497

This text gives a basic introduction, and a unified approach, to algebra and geometry. Alan Beardon covers the ideas of complex numbers, scalar and vector products, determinants, linear algebra, group theory, permutation groups, symmetry groups, and various aspects of geometry including groups of isometries, rotations, and spherical geometry. The emphasis is on the interaction among these topics. The text is divided into short sections, with exercises at the end of each section.

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Table of contents :
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
Preface……Page 13
1.1 Introduction……Page 15
1.2 Groups……Page 16
Exercise 1.2……Page 19
1.3 Permutations of a finite set……Page 20
1.4 The sign of a permutation……Page 25
1.5 Permutations of an arbitrary set……Page 29
Exercise 1.5……Page 34
2.1 The integers……Page 36
2.2 The real numbers……Page 40
2.3 Fields……Page 41
2.4 Modular arithmetic……Page 42
Exercise 2.4……Page 43
3.1 Complex numbers……Page 45
3.2 Polar coordinates……Page 50
3.3 Lines and circles……Page 54
3.4 Isometries of the plane……Page 55
Exercise 3.4……Page 57
3.5 Roots of unity……Page 58
3.6 Cubic and quartic equations……Page 60
Exercise 3.6……Page 61
3.7 The Fundamental Theorem of Algebra……Page 62
Exercise 3.7……Page 65
4.1 Vectors……Page 66
Exercise 4.1……Page 68
4.2 The scalar product……Page 69
4.3 The vector product……Page 71
4.4 The scalar triple product……Page 74
4.5 The vector triple product……Page 76
4.6 Orientation and determinants……Page 77
(A) The geometry of lines……Page 82
(B) The geometry of planes……Page 83
(C) The geometry of triangles……Page 84
Exercise 4.7……Page 85
4.8 Vector equations……Page 86
Exercise 4.8……Page 87
5.1 Spherical distance……Page 88
5.2 Spherical trigonometry……Page 89
5.3 Area on the sphere……Page 91
5.4 Euler’s formula……Page 93
5.5 Regular polyhedra……Page 97
Exercise 5.5……Page 98
5.6 General polyhedra……Page 99
Exercise 5.6……Page 102
6.1 Isometries of Euclidean space……Page 103
6.2 Quaternions……Page 109
Exercise 6.2……Page 112
6.3 Reflections and rotations……Page 113
Exercise 6.3……Page 114
7.1 Vector spaces……Page 116
Exercise 7.1……Page 119
7.2 Dimension……Page 120
Exercise 7.2……Page 124
7.3 Subspaces……Page 125
7.4 The direct sum of two subspaces……Page 129
Exercise 7.4……Page 131
7.5 Linear difference equations……Page 132
7.6 The vector space of polynomials……Page 134
7.7 Linear transformations……Page 138
Exercise 7.7……Page 140
7.8 The kernel of a linear transformation……Page 141
Exercise 7.8……Page 143
7.9 Isomorphisms……Page 144
7.10 The space of linear maps……Page 146
Exercise 7.10……Page 148
8.1 Hyperplanes……Page 149
8.2 Homogeneous linear equations……Page 150
8.3 Row rank and column rank……Page 153
Exercise 8.3……Page 154
8.4 Inhomogeneous linear equations……Page 155
8.5 Determinants and linear equations……Page 157
8.6 Determinants……Page 158
Exercise 8.6……Page 162
9.1 The vector space of matrices……Page 163
Exercise 9.1……Page 167
9.2 A matrix as a linear transformation……Page 168
Exercise 9.2……Page 171
9.3 The matrix of a linear transformation……Page 172
9.4 Inverse maps and matrices……Page 177
9.5 Change of bases……Page 181
Exercise 9.5……Page 183
9.6 The resultant of two polynomials……Page 184
9.7 The number of surjections……Page 187
Exercise 9.7……Page 188
10.1 Eigenvalues and eigenvectors……Page 189
Exercise 10.1……Page 193
10.2 Eigenvalues and matrices……Page 194
Exercise 10.2……Page 197
10.3 Diagonalizable matrices……Page 198
Exercise 10.3……Page 202
10.4 The Cayley–Hamilton theorem……Page 203
Exercise 10.4……Page 206
10.5 Invariant planes……Page 207
Exercise 10.5……Page 209
11.1 Distance in Euclidean space……Page 211
11.2 Orthogonal maps……Page 212
Exercise 11.2……Page 217
11.3 Isometries of Euclidean n-space……Page 218
11.4 Symmetric matrices……Page 220
Exercise 11.4……Page 224
11.5 The field axioms……Page 225
11.6 Vector products in higher dimensions……Page 226
Exercise 11.6……Page 228
12.1 Groups……Page 229
12.2 Subgroups and cosets……Page 232
Exercise 12.2……Page 236
12.3 Lagrange’s theorem……Page 237
12.4 Isomorphisms……Page 239
Exercise 12.4……Page 243
12.5 Cyclic groups……Page 244
12.6 Applications to arithmetic……Page 246
Exercise 12.6……Page 248
12.7 Product groups……Page 249
Exercise 12.7……Page 250
12.8 Dihedral groups……Page 251
Exercise 12.8……Page 253
12.9 Groups of small order……Page 254
Exercise 12.9……Page 255
12.10 Conjugation……Page 256
Exercise 12.10……Page 259
12.11 Homomorphisms……Page 260
12.12 Quotient groups……Page 263
Exercise 12.12……Page 267
13.1 Möbius transformations……Page 268
Exercise 13.1……Page 272
13.2 Fixed points and uniqueness……Page 273
13.3 Circles and lines……Page 275
13.4 Cross-ratios……Page 279
13.5 Möbius maps and permutations……Page 282
13.6 Complex lines……Page 285
13.7 Fixed points and eigenvectors……Page 287
Exercise 13.7……Page 289
13.8 A geometric view of infinity……Page 290
Exercise 13.8……Page 292
13.9 Rotations of the sphere……Page 293
Exercise 13.9……Page 296
14.1 Groups of permutations……Page 298
Exercise 14.1……Page 303
14.2 Symmetries of a regular polyhedron……Page 304
14.3 Finite rotation groups in space……Page 309
14.4 Groups of isometries of the plane……Page 311
14.5 Group actions……Page 317
Exercise 14.5……Page 320
15.1 The hyperbolic plane……Page 321
15.2 The hyperbolic distance……Page 324
15.3 Hyperbolic circles……Page 327
Exercise 15.3……Page 328
15.4 Hyperbolic trigonometry……Page 329
15.5 Hyperbolic three-dimensional space……Page 331
15.6 Finite Möbius groups……Page 333
Index……Page 334