Book: G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number


New Foundations in Mathematics: The Geometric Concept of Number

by Garret Sobczyk, Universidad de las Am´ericas-Puebla, Departamento de F´ısico-Matem´aticas, Apartado Postal #100, Santa Catarina M´artir, 72820 Puebla, Pue., M´exico, garret_sobczykATyahoo.com
©2009 by Garret Sobczyk, August 16, 2010, 273 pages

Downlaod: http://www.garretstar.com/NFM16VIII10.pdf

Preface
The development of the real number system, and the identification of real numbers with points on the real number line, represents both a milestone and a cornerstone in the foundation of modern mathematics. We assume readers are familiar with the real and complex number systems. By a field we mean the real or complex numbers, but we also consider modular number systems. …

Contents
1 Modular Number Systems 11
1.1 Beginnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Modular Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Modular Polynomials . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Interpolation Polynomials . . . . . . . . . . . . . . . . . . . . 22
1.5 Generalized Taylor’s Theorem . . . . . . . . . . . . . . . . . . 26
2 Complex and Hyperbolic Numbers 33
2.1 The hyperbolic numbers . . . . . . . . . . . . . . . . . . . . . 34
2.2 Hyperbolic Polar Form . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Inner and outer products . . . . . . . . . . . . . . . . . . . . . 41
2.4 Idempotent basis . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 The cubic equation . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6 Special relativity and Lorentzian geometry . . . . . . . . . . . 49
3 Geometric Algebra 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Geometric numbers of the plane . . . . . . . . . . . . . . . . 54
3.3 The geometric algebra IG3 of space . . . . . . . . . . . . . . . 59
3.4 Orthogonal transformations . . . . . . . . . . . . . . . . . . . 63
3.5 Geometric algebra of IRn . . . . . . . . . . . . . . . . . . . . . 66
3.6 Vector derivative in IRn . . . . . . . . . . . . . . . . . . . . . . 70
4 Vector Spaces and Matrices 75
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Matrix multiplication . . . . . . . . . . . . . . . . . . . 81
4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Rules of matrix algebra . . . . . . . . . . . . . . . . . . . . . . 85
4.4 The matrices of IG2 and IG3 . . . . . . . . . . . . . . . . . . . . 85
4.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Outer Product and Determinants 91
5.1 The outer product . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Applications to matrices . . . . . . . . . . . . . . . . . . . . . 98
5.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Systems of Linear Equations 101
6.1 Elementary operations and matrices . . . . . . . . . . . . . . . 101
6.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Gauss Jordan Elimination . . . . . . . . . . . . . . . . . . . . 106
6.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 LU decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7 Linear Transformations on IRn 113
7.1 Definition of a linear transformation . . . . . . . . . . . . . . . 113
7.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 The adjoint tranformation . . . . . . . . . . . . . . . . . . . . 118
7.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8 Structure of a Linear Operator 121
8.1 Rank of a linear operator . . . . . . . . . . . . . . . . . . . . . 121
8.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.2 Characteristic polynomial . . . . . . . . . . . . . . . . . . . . 124
8.2.1 Minimal polynomial of f . . . . . . . . . . . . . . . . . 126
8.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.3 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . 128
8.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.4 Jordan normal form . . . . . . . . . . . . . . . . . . . . . . . . 133
8.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9 Linear and Bilinear Forms 141
9.1 The Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.4 The Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.5 Hermitian forms . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10 Hermitian Inner Product Spaces 161
10.1 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . 161
10.2 Orthogonality relationships in euclidean space . . . . . . . . . 166
10.3 Unitary geometric algebra of pseudoeucliean space . . . . . . . 168
10.4 Hermitian orthogonality . . . . . . . . . . . . . . . . . . . . . 173
10.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10.5 Hermitian, Normal, and Unitary Operators . . . . . . . . . . . 179
10.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 182
10.6 Principal Correlation . . . . . . . . . . . . . . . . . . . . . . . 183
10.7 Polar and Singular Value Decomposition . . . . . . . . . . . . 186
11 Geometry of moving planes 189
11.1 Relative geometric algebras . . . . . . . . . . . . . . . . . . . 191
11.2 Moving planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
11.3 Splitting the plane . . . . . . . . . . . . . . . . . . . . . . . . 197
12 Representation of the Symmetric Group 205
12.1 The twisted symmetric product . . . . . . . . . . . . . . . . . 205
12.1.1 Special properties . . . . . . . . . . . . . . . . . . . . . 207
12.1.2 Basic relationships . . . . . . . . . . . . . . . . . . . . 208
12.2 Geometric numbers in IG = cln,n . . . . . . . . . . . . . . . . . 209
12.3 The twisted symmetric product of geometric numbers . . . . . 211
12.4 Symmetric groups in geometric algebras . . . . . . . . . . . . 216
12.4.1 The symmetric group S4 in cl4,4 . . . . . . . . . . . . . 216
12.4.2 The geometric algebra IG = cl4,4 . . . . . . . . . . . . . 220
12.4.3 The general construction in cln,n . . . . . . . . . . . . . 223
12.5 The heart of the matter . . . . . . . . . . . . . . . . . . . . . 224
13 Calculus on k-Surfaces 229
13.1 The boundary of a surface . . . . . . . . . . . . . . . . . . . . 229
13.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 232
13.2 The directed integral . . . . . . . . . . . . . . . . . . . . . . . 234
13.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 237
13.3 Classical theorems of integration . . . . . . . . . . . . . . . . . 238
13.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 239
14 Differential Geometry of Curves 241
14.1 Definition of a curve . . . . . . . . . . . . . . . . . . . . . . . 241
14.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 242
14.2 Formulas of Frenet-Serret . . . . . . . . . . . . . . . . . . . . 243
14.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 245
14.3 Special curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
14.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 247
14.4 Uniqueness theorem for curves . . . . . . . . . . . . . . . . . . 248
14.4.1 Exercises: . . . . . . . . . . . . . . . . . . . . . . . . . 248
15 Differential Geometry of k-Surfaces 253
15.1 The metric tensor of M. . . . . . . . . . . . . . . . . . . . . . 253
15.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 256
15.2 Second Fundamental Form . . . . . . . . . . . . . . . . . . . . 257
15.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 259
15.3 Unit speed curves on M . . . . . . . . . . . . . . . . . . . . . 260
15.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 263
15.4 Gaussian, Mean, and Principal Curvatures of M. . . . . . . . 263
15.5 The Curvature Tensor of a Surface M. . . . . . . . . . . . . . 264
15.6 Isometries and Rigid Motions . . . . . . . . . . . . . . . . . . 266
15.7 Affine connections . . . . . . . . . . . . . . . . . . . . . . . . . 267

Source: http://www.garretstar.com/NFM16VIII10.pdf

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