1 FOUNDATIONS OF COMPUTATION 1
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| 1.1 Introduction | 1
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| 1.2 Representations of Numbers | 2
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| 1.2.1 Integers | 3
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| 1.2.2 Rational Numbers and Real Numbers | 14
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| 1.2.3 Representations of Numbers as Text | 17
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| 1.2.4 Exercises for Section 1.2 | 20
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| 1.3 Finite Floating-point Representations | 21
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| 1.3.1 Simple Cases | 21
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| 1.3.2 Practical Floating-point Representations | 25
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| 1.3.3 Approaching Zero or In.nity Gracefully | 28
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| 1.3.4 Exercises for Section 1.3 | 30
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| 1.4 Floating-point Computation | 31
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| 1.4.1 Relative Error; Machine Epsilon | 31
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| 1.4.2 Rounding | 32
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| 1.4.3 Floating-point Addition and Subtraction | 35
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| 1.4.4 Exercises for Section 1.4 | 36
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| 1.5 Propagation of Errors | 37
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| 1.5.1 General Formulas | 37
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| 1.5.2 Examples of Error Propagation | 39
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| 1.5.3 Estimates of the Mean and Variance | 41
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| 1.5.4 Exercises for Section 1.5 | 43
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| 1.6 Bibliography and Endnotes | 45
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| 1.6.1 Bibliography | 45
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| 1.6.2 Endnotes | 46
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2 SETS AND MAPPINGS 47
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| 2.1 Introduction | 47
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| 2.2 Basic De.nitions | 49
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| 2.2.1 Sets | 49
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| 2.2.2 Mappings | 53
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| 2.2.3 Axiom of Choice | 62
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| 2.2.4 Cartesian Products | 62
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| 2.2.5 Equivalence and Equivalence Classes | 65
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| 2.2.6 Exercises for Section 2.2 | 67
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| 2.3 Union, Intersection and Complement | 68
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| 2.3.1 Unions of Sets | 68
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| 2.3.2 Intersections of Sets | 69
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| 2.3.3 The Relative Complement | 70
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| 2.3.4 De Morgan’s Laws | 71
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| 2.3.5 Exercises for Section 2.3 | 71
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| 2.4 In.nite Sets | 72
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| 2.4.1 Basic Properties of In.nite Sets | 72
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| 2.4.2 Induction and Recursion | 73
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| 2.4.3 Countable Sets | 76
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| 2.4.4 Countable Unions and Intersections | 77
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| 2.4.5 Uncountable Sets | 78
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| 2.4.6 Exercises for Section 2.4 | 80
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| 2.5 Ordered and Partially Ordered Sets | 82
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| 2.5.1 Partial Orderings | 82
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| 2.5.2 Orderings; Upper and Lower Bounds | 83
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| 2.5.3 Maximal Chains | 84
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| 2.5.4 Exercises for Section 2.5 | 84
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| 2.6 Bibliography | 85
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3 EVALUATION OF FUNCTIONS 86
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| 3.1 Introduction | 86
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| 3.2 Sensitivity and Condition Number | 86
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| 3.2.1 De.nitions | 86
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| 3.2.2 Evaluation of Polynomials | 87
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| 3.2.3 Multiple Roots of Polynomials | 89
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| 3.2.4 Exercises for Section 3.2 | 91
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| 3.3 Recursion and Iteration | 92
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| 3.3.1 Finding Roots by Bisection | 92
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| 3.3.2 The Newton-Raphson Method | 92
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| 3.3.3 Evaluation of Series | 95
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| 3.3.4 Exercises for Section 3.3 | 97
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| 3.4 Introduction to Numerical Integration | 99
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| 3.4.1 Rectangle Rules | 101
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| 3.4.2 Trapezoidal Rule | 102
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| 3.4.3 Local and Global Errors | 102
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| 3.4.4 Exercises for Section 3.4 | 105
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| 3.5 Solution of Di.erential Equations | 106
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3 CONTENTS
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| 3.5.1 Euler’s Method | 107
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| 3.5.2 Truncation Error of Euler’s Method | 109
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| 3.5.3 Stability Analysis of Euler’s Method | 111
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| 3.5.4 Selected Finite-Di.erence Methods | 112
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| 3.5.5 Exercises for Section 3.5 | 118
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| 3.6 Bibliography | 120
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4 GROUPS, RINGS AND FIELDS 121
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| 4.1 Introduction | 121
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| 4.2 Groups | 122
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| 4.2.1 Axioms | 122
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| 4.2.2 Two-Element Group | 127
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| 4.2.3 Orbits and Cosets | 130
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| 4.2.4 Cyclic Groups | 136
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| 4.2.5 Dihedral Groups | 141
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| 4.2.6 Cubic Groups | 143
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| 4.2.7 Continuous Groups | 143
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| 4.2.8 Classes of Conjugate Elements | 147
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| 4.2.9 Exercises for Section 4.2 | 149
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| 4.3 Group Homomorphisms | 152
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| 4.3.1 De.nitions and Basic Properties | 152
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| 4.3.2 Normal Subgroups | 158
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| 4.3.3 Direct Product Groups | 162
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| 4.3.4 Exercises for Section 4.3 | 164
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| 4.4 *Symmetric Groups | 165
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| 4.4.1 Permutations | 165
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| 4.4.2 Cayley’s Theorem | 167
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| 4.4.3 Cyclic Permutations | 169
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| 4.4.4 Even and Odd Permutations | 171
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| 4.4.5 Exercises for Section 4.4 | 173
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| 4.5 Rings and Integral Domains | 175
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| 4.5.1 Axioms and Examples | 175
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| 4.5.2 Basic Properties of Rings | 179
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| 4.5.3 Rational Numbers | 180
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| 4.5.4 *Ring Homomorphisms | 181
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| 4.5.5 Exercises for Section 4.5 | 184
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| 4.6 Fields | 184
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| 4.6.1 Axioms and Examples | 184
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| 4.6.2 *Galois Fields | 186
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| 4.6.3 Exercises for Section 4.6 | 189
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| 4.7 Bibliography | 189
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5 VECTOR SPACES 191
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| 5.1 Introduction | 191
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| 5.2 Basic De.nitions and Examples | 193
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| 5.2.1 Axioms for a Vector Space | 193
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| 5.2.2 Selected Realizations of the Vector-Space Axioms | 194
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| 5.2.3 Vector Subspaces | 201
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| 5.2.4 *Comments on Vector-Space Axioms | 205
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| 5.2.5 Exercises for Section 5.2 | 208
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| 5.3 Linear Independence and Linear Dependence | 211
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| 5.3.1 De.nitions | 211
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| 5.3.2 Basic results on Linear Dependence | 212
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| 5.3.3 Examples of Linear Independence | 216
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| 5.3.4 Exercises for Section 5.3 | 219
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| 5.4 Bases and Dimension | 221
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| 5.4.1 Dimension of a Vector Space | 221
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| 5.4.2 Selected Realizations of Vector-Space Bases | 225
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| 5.4.3 Vector-Space Isomorphisms | 228
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| 5.4.4 Gaussian Elimination and Linear Dependence | 232
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| 5.4.5 Exercises for Section 5.4 | 237
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| 5.5 Complementary Subspaces | 239
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| 5.5.1 Vector Complements and Direct Sums | 239
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| 5.5.2 De.nition of Complementary Subspaces | 240
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| 5.5.3 Dimensions of Complementary Subspaces | 241
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| 5.5.4 Direct Sums of Vector Spaces | 242
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| 5.5.5 Bases of Complementary Subspaces | 243
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| 5.5.6 Examples of Direct Sums of Vector Spaces | 244
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| 5.5.7 Exercises for Section 5.5 | 245
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| 5.6 Bibliography and Endnotes | 246
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| 5.6.1 Bibliography | 246
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| 5.6.2 Endnotes | 246
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6 LINEAR MAPPINGS I 248
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| 6.1 Linear Mappings and their Matrices | 248
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| 6.1.1 Basic Properties | 248
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| 6.1.2 Matrix of a Linear Mapping | 250
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| 6.1.3 Computation of Matrix Products | 261
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| 6.1.4 Invariant Subspaces and Direct Sums | 263
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| 6.1.5 Other examples of Linear Mappings | 265
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| 6.1.6 Exercises for Section 6.1 | 268
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| 6.2 Nonsingular Linear Mappings | 271
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| 6.2.1 De.nitions and Basic Properties | 271
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| 6.2.2 Change of Basis | 275
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| 6.2.3 Permutation Matrices | 277
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| 6.2.4 General Linear Group of a Vector Space | 278
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| 6.2.5 Exercises for Section 6.2 | 278
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| 6.3 Singular Linear Mappings | 281
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| 6.3.1 Singularity and Linear Dependence | 281
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| 6.3.2 Visualization of Singular Linear Mappings | 282
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| 6.3.3 Null space of a linear mapping | 283
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| 6.3.4 Other Examples of Singular Linear Mappings | 285
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| 6.3.5 Exercises for Section 6.3 | 287
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| 6.4 Introduction to Digital Filters | 288
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| 6.4.1 De.nitions | 288
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| 6.4.2 Noise Ampli.cation by Digital Filters | 291
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| 6.4.3 Di.erence Operators | 292
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| 6.4.4 Exercises for Section 6.4 | 298
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| 6.5 Trace and Determinant | 299
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| 6.5.1 Trace of a Linear Mapping | 299
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| 6.5.2 Determinants | 300
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| 6.5.3 Exercises for Section 6.5 | 309
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| 6.6 Solution of Linear Equations | 310
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| 6.6.1 Basic Facts about Linear Equations | 310
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| 6.6.2 Matrix Formulation of Gaussian Elimination | 312
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| 6.6.3 Computational Aspects of Gaussian Elimination | 317
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| 6.6.4 The LU and LDMT Decompositions | 317
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| 6.6.5 Bases of the Range and Null Space | 319
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| 6.6.6 Rank-nullity Theorem | 321
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| 6.6.7 Exercises for Section 6.6 | 322
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| 6.7 Complements of Null Space | 324
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| 6.7.1 Quotient Space V/null [A] | 324
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| 6.7.2 Isomorphism of the Range to a Complement of the Null Space | 325
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| 6.7.3 Rank-nullity Theorem (Again) | 327
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| 6.7.4 Right Inverses of a Linear Mapping | 327
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| 6.7.5 Examples of Right Inverses | 328
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| 6.7.6 Exercises for Section 6.7 | 330
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| 6.8 Bibliography | 330
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7 LINEAR FUNCTIONALS 331
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| 7.1 Motivation for Studying Functionals | 331
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| 7.2 Dual Spaces | 332
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| 7.2.1 De.nitions | 332
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| 7.2.2 Range and Null Space of a Linear Functional | 334
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| 7.2.3 Exercises for Section 7.2 | 335
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| 7.3 Coordinate Functionals | 336
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| 7.3.1 De.nitions | 336
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| 7.3.2 Coordinate Functionals on Fn | 337
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| 7.3.3 Isomorphism ofV* to V | 338
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| 7.3.4 Coordinate Functionals on Two-Dimensional Euclidean Space | 339
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| 7.3.5 Coordinate Functionals and the Reciprocal Lattice | 341
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| 7.3.6 Isomorphism of V to V** | 345
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| 7.3.7 Exercises for Section 7.3 | 346
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| 7.4 Annihilator of a Subspace | 347
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| 7.4.1 De.nitions | 347
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| 7.4.2 Bases of the Annihilator | 347
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| 7.4.3 Exercises for Section 7.4 | 348
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| 7.5 Other Realizations of Dual Spaces | 349
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| 7.5.1 Dual space of C | 349
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| 7.5.2 Dual of FZ+ | 349
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| 7.5.3 Boundary and Initial Conditions for Di.erential Equations | 349
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| 7.6 Polynomial Interpolation | 350
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| 7.6.1 Lagrangian Interpolation | 350
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| 7.6.2 Exercises for Section 7.6 | 352
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| 7.7 Tensors | 352
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| 7.7.1 De.nitions and Basic Properties | 353
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| 7.7.2 Components of Second-Rank Tensors | 356
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| 7.7.3 Tensor Products of Vectors | 358
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| 7.7.4 Tensors of Rank m | 361
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| 7.7.5 Linear Mappings of Tensors | 362
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| 7.7.6 Exercises for Section 7.7 | 365
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8 INNER PRODUCTS AND NORMS 367
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| 8.1 Inner-product spaces | 367
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| 8.1.1 De.nitions | 367
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| 8.1.2 Canonical Inner Products | 369
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| 8.1.3 Metric Tensor | 372
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| 8.1.4 Inde.nite Inner Products | 378
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| 8.1.5 Orthogonality | 379
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| 8.1.6 Exercises for Section 8.1 | 383
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| 8.2 Geometry of Inner-Product Spaces | 384
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| 8.2.1 Pythagoras’s Theorem | 384
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| 8.2.2 Orthonormal Bases | 392
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| 8.2.3 Orthogonal Polynomials | 397
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| 8.2.4 Exercises for Section 8.2 | 403
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| 8.3 Projection Methods | 406
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| 8.3.1 Projection of a Vector onto a Subspace: De.nition | 406
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| 8.3.2 Orthogonal Projectors | 407
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| 8.3.3 Orthogonal Complement | 409
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| 8.3.4 Exercises for Section 8.3 | 416
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| 8.4 Least-squares Approximations | 417
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| 8.4.1 Motivation | 417
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| 8.4.2 Abstract Formulation | 418
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| 8.4.3 Inequalities for Least-squares Approximations | 419
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| 8.4.4 Approximation by Finite Fourier Sums | 420
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| 8.4.5 Chebyshev Approximations | 421
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| 8.4.6 Mapping a Function to its Fourier Coe.cients | 422
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| 8.4.7 Exercises for Section 8.4 | 423
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| 8.5 Discrete Fourier Transform | 424
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| 8.5.1 Approximation of Fourier Coe.cients | 424
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| 8.5.2 Discrete Fourier Basis | 425
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| 8.5.3 Periodic Extension | 428
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| 8.5.4 Aliasing | 430
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| 8.5.5 Sampling Theorem and Alias Mapping | 433
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| 8.5.6 Exercises for Section 8.5 | 437
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| 8.6 Volume of an m-Parallelepiped | 437
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| 8.6.1 Parallelepipeds | 437
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| 8.6.2 Recursive De.nition of Volume | 438
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| 8.6.3 Volume as a Determinant | 438
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| 8.6.4 Determinant as a Volume Ratio | 440
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| 8.6.5 Jacobian Determinant | 441
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| 8.6.6 Exercises for Section 8.6 | 443
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| 8.7 Vector and Matrix Norms | 443
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| 8.7.1 Vector Norms | 443
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| 8.7.2 Norm of a Linear Mapping | 446
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| 8.7.3 Matrix Norms | 450
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| 8.7.4 Norm of an Integral | 453
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| 8.7.5 Exercises for Section 8.7 | 453
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| 8.8 Inner Products and Linear Functionals | 455
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| 8.8.1 Introduction | 455
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| 8.8.2 Inner-product Mapping | 456
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| 8.8.3 Inverse Inner-product Mapping | 459
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| 8.8.4 Exercises for Section 8.8 | 464
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| 8.9 Bibliography and Endnotes | 464
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| 8.9.1 Bibliography | 465
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| 8.9.2 Endnotes | 465
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9 LINEAR MAPPINGS II 466
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| 9.1 Dyads | 466
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| 9.1.1 Motivation | 466
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| 9.1.2 De.nition of a Dyad | 466
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| 9.1.3 Dyadic Expansions | 469
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| 9.1.4 Resolutions of the Identity Mapping | 471
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| 9.1.5 Exercises for Section 9.1 | 473
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| 9.2 Transpose and Adjoint | 473
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| 9.2.1 Transpose | 473
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| 9.2.2 Adjoint | 476
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| 9.2.3 Other Realizations of the Adjoint | 480
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| 9.2.4 Properties of the Adjoint | 483
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| 9.2.5 Hermitian and Self-adjoint Mappings | 485
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| 9.2.6 Isometric and Unitary Mappings | 487
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| 9.2.7 Exercises for Section 9.2 | 491
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| 9.3 Eigenvalues and Eigenvectors | 493
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| 9.3.1 Secular Equation | 493
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| 9.3.2 Diagonalization of Hermitian Matrices | 495
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| 9.3.3 Normal Linear Mappings | 502
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| 9.3.4 Exercises for Section 9.3 | 504
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| 9.4 Singular-Value Decomposition | 507
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| 9.4.1 Derivation of the Singular-Value Decomposition | 507
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| 9.4.2 Matrix Version of the Singular-Value Decomposition | 509
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| 9.4.3 The Fundamental Subspaces of a Linear Mapping | 511
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| 9.4.4 Inverse and Pseudo-inverse in the SVD | 512
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| 9.4.5 Data Compression Using the SVD | 514
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| 9.4.6 Exercises for Section 9.4 | 514
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| 9.5 Linear Equations II | 515
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| 9.5.1 Numerical Versus Analytical Methods | 515
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| 9.5.2 Diagonal Dominance | 516
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| 9.5.3 Condition Number of the Linear-equation Problem | 517
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| 9.5.4 The LDL† and Cholesky decompositions | 520
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| 9.6 Selected Applications of Linear Equations | 521
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| 9.6.1 The Linear Least-squares Problem | 521
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| 9.6.2 Linear Di.erence Equations | 523
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| 9.6.3 Solution of Tridiagonal Systems | 529
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| 9.6.4 Exercises for Section 9.6 | 530
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| 9.7 Bibliography | 531
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10 CONVERGENCE IN NORMED VECTOR SPACES 532
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| 10.1 Metrics and Norms | 532
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| 10.1.1 Metric Spaces | 532
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| 10.1.2 Normed Vector Spaces | 534
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| 10.1.3 Examples of Metric and Normed Vector Spaces | 536
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| 10.1.4 Open Sets | 539
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| 10.1.5 Exercises for Section 10.1 | 541
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| 10.2 Limit Points | 543
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| 10.2.1 Limit Points and Closed Sets | 543
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| 10.2.2 Dense Sets and Separable Spaces | 548
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| 10.2.3 Exercises for Section 10.2 | 552
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| 10.3 Convergence of Sequences and Series | 553
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| 10.3.1 Convergence of Sequences | 553
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| 10.3.2 Numerical Sequences | 558
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| 10.3.3 Numerical Series | 560
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| 10.3.4 Exercises for Section 10.3 | 565
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| 10.4 Strong and Pointwise Convergence | 566
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| 10.4.1 Strong Convergence | 566
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| 10.4.2 Operators | 570
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| 10.4.3 Sequences of Real-valued Functions | 572
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| 10.4.4 Series of Real-valued Functions | 575
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| 10.4.5 Exercises for Section 10.4 | 576
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| 10.5 Continuity | 577
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| 10.5.1 Pointwise Continuity | 577
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| 10.5.2 Uniform Continuity | 580
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| 10.6 Best Approximations in the Maximum and Supremum Norms | 581
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| 10.6.1 Best Approximations in the Maximum Norm | 583
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| 10.6.2 Best Approximations in the Supremum Norm | 589
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| 10.6.3 Exercises for Section 10.6 | 592
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| 10.7 Hilbert and Banach Spaces | 594
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| 10.7.1 Survey of Complete Metric Vector Spaces | 594
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| 10.7.2 Complete Orthonormal Sets | 597
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| 10.7.3 Orthogonal Series | 599
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| 10.7.4 Practical Aspects of Fourier Series | 603
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| 10.7.5 Orthogonal-polynomial Expansions | 610
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| 10.7.6 Exercises for Section 10.7 | 613
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| 10.8 Bibliography | 615
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11 GROUP REPRESENTATIONS 616
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| 11.1 Preliminaries | 616
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| 11.1.1 Background | 616
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| 11.1.2 Symmetry-adapted Functions | 618
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| 11.1.3 Partner Functions | 619
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| 11.1.4 Exercises for Section 11.1 | 621
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| 11.2 Reducibility of Representations | 621
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| 11.2.1 Invariant Subspaces and Irreducibility | 622
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| 11.2.2 Schur’s Lemma | 623
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| 11.2.3 Eigenvectors of Invariant Operators | 627
|
| 11.2.4 Exercises for Section 11.2 | 629
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| 11.3 Unitarity and Orthogonality | 630
|
| 11.3.1 Consequences of the Rearrangement Theorem | 630
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| 11.3.2 Unitary Representations | 631
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| 11.3.3 Orthogonality Theorems | 633
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| 11.3.4 Product Relation for Characters | 638
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| 11.3.5 Reduction of Unitary Representations | 640
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| 11.3.6 Construction of Character Tables | 643
|
| 11.3.7 Characters of Kronecker Products | 644
|
| 11.3.8 Exercises for Section 11.3 | 646
|
| 11.4 Two-dimensional rotation group | 647
|
| 11.4.1 Representation space for SO(2) | 648
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| 11.4.2 Representations of SO(2) | 649
|
| 11.4.3 Completeness Relation for e.imè | 650
|
| 11.4.4 Exercise for Section 11.4 | 652
|
| 11.5 Symmetry and the One-dimensional Wave Equation | 652
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| 11.5.1 Boundary Conditions and Symmetry | 652
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| 11.5.2 Wave Equation for a Vibrating String | 653
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| 11.5.3 Boundary Conditions for the One-dimensional Wave Equation | 653
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| 11.5.4 Form Invariance of the Wave Equation | 653
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| 11.5.5 Invariance of the Wave Equation under Translations | 656
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| 11.5.6 Invariance of the Wave Equation under Lorentz Transformations | 656
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| 11.5.7 D’Alembert’s Solution of the Wave Equation | 657
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| 11.5.8 Solution for a String of In.nite Length | 658
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| 11.5.9 Solution for a String of Finite Length | 659
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| 11.5.10 Exercises for Section 11.5 | 661
|
| 11.6 Discrete Translation Groups | 662
|
| 11.6.1 Motivation | 662
|
| 11.6.2 Invariance Under the Discrete Translation Group | 662
|
| 11.6.3 Discrete-shift-invariant Digital Filters | 663
|
| 11.6.4 Representations of the Discrete Translation Group | 664
|
| 11.6.5 Discrete-time Transfer Function | 665
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| 11.6.6 Exercises for Section 11.6 | 668
|
| 11.7 Continuous Translation Groups | 669
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| 11.7.1 Translation Group of the Real Line | 669
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| 11.7.2 Irreducible Representations of T (R) | 669
|
| 11.7.3 ø as Momentum Eigenfunctions | 671
|
| 11.7.4 Representation of T(E2) Carried by øk | 672
|
| 11.7.5 Translation Group of Euclidean n-space | 673
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| 11.7.6 Exercises for Section 11.7 | 676
|
| 11.8 Fourier transforms | 676
|
| 11.8.1 Fourier Transform in One Dimension | 676
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| 11.8.2 Completeness Relation for the øk | 677
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| 11.8.3 Fourier Transforms in n-dimensional Euclidean Space | 678
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| 11.8.4 Poisson sum formula | 679
|
| 11.8.5 Exercises for Section 11.8 | 679
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| 11.9 Linear, Shift-invariant Systems | 680
|
| 11.9.1 Continuous-time-shift Invariance | 680
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| 11.9.2 Continuous-time Transfer Function | 681
|
| 11.9.3 Exercises for Section 11.9 | 682
|
| 11.10Two-dimensional Euclidean Group E(2) | 682
|
| 11.10.1 Representation of E(2) Carried by øk | 684
|
| 11.10.2 Discussion | 684
|
| 11.10.3 Exercises for Section 11.10 | 685
|
| 11.11Bibliography | 685
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12 SPECIAL FUNCTIONS 686
|
| 12.1 Group Theory and Special Functions | 686
|
| 12.1.1 Separation of Variables | 686
|
| 12.1.2 Special Functions as Matrix Elements | 687
|
| 12.1.3 Symmetries of the Helmholtz Equation | 687
|
| 12.1.4 Exercise for Section 12.1 | 690
|
| 12.2 De.nition of the Bessel Functions | 690
|
| 12.2.1 Fourier Expansion of a Plane Wave | 690
|
| 12.2.2 De.nition of the Bessel Function Jm of Integer Order | 692
|
| 12.2.3 Jacobi-Anger expansion | 692
|
| 12.2.4 Frequency Content of an FM Signal | 694
|
| 12.2.5 Exercises for Section 12.2 | 694
|
| 12.3 Bessel-function addition formulas | 695
|
| 12.3.1 Exercises for Section 12.3 | 698
|
| 12.4 Bessel Raising and Lowering Operators | 698
|
| 12.4.1 Recurrence Relations | 698
|
| 12.4.2 Raising and Lowering Operators for Jm | 700
|
| 12.4.3 Raising and Lowering Operators for the Helmholtz functions | 700
|
| 12.4.4 Exercises for Section 12.4 | 701
|
| 12.5 Bessel Di.erential Equations | 701
|
| 12.5.1 Bessel’s Di.erential Equation | 701
|
| 12.5.2 The Helmholtz Equation in Two Dimensions | 702
|
| 12.5.3 Qualitative Behavior of Jm | 702
|
| 12.5.4 Exercises for Section 12.5 | 704
|
| 12.6 Orthogonal Series in Jn | 704
|
| 12.6.1 Boundary Conditions that Ensure Self-adjointness | 704
|
| 12.6.2 Orthogonality Relations | 708
|
| 12.6.3 Fourier-Bessel Series | 709
|
| 12.6.4 Exercise for Section 12.6 | 711
|
| 12.7 Vibrations of a Drumhead | 711
|
| 12.7.1 Exercise for Section 12.7 | 713
|
| 12.8 Power Series for Jm | 713
|
| 12.8.1 Derivation using Raising and Lowering Operators | 713
|
| 12.8.2 Properties Deduced from the Power Series | 715
|
| 12.8.3 Exercises for Section 12.8 | 715
|
| 12.9 Completeness Relations using Jn | 716
|
| 12.10Bibliography | 718
|
A INDEX OF NOTATION 719
|
| A.1 Quanti.ers and Other Logical Symbols | 719
|
| A.2 Sets and Mappings | 719
|
| A.3 Vector Spaces, Linear Mappings and Matrices | 720
|
| A.4 Norms and Inner Products | 721
|
| A.5 Functions | 721
|
| A.5.1 General Notation for Functions | 721
|
| A.5.2 Special Functions | 722
|
| A.6 Probability | 722
|
B AFFINE MAPPINGS 723
|
| B.1 A.ne Group of a Vector Space | 723
|
| B.2 Coordinate Transformations | 726
|
| B.2.1 Active Transformations | 727
|
| B.2.2 Passive Transformations | 727
|
| B.2.3 Relation between Active and Passive Transformations | 728
|
| B.3 Exercises | 728
|
C PSEUDO-UNITARY SPACES 730
|
D REMAINDER TERM 733
|
E BOLZANO-WEIERSTRASS THEOREM 735
|
| E.1 Real Numbers | 735
|
| E.2 Finite-dimensional Hilbert Spaces | 736
|
F WEIERSTRASS APPROXIMATION THEOREM 738
|
