A Programmer’s Introduction to Mathematics

Contents......Page 3
Our Goal......Page 6
Mathematics has a Culture......Page 9
Polynomials, Java, and Definition......Page 13
A LittleMore Notation......Page 21
Existence & Uniqueness......Page 22
Realizing it in Code......Page 30
Application: Sharing Secrets......Page 32
Cultural Review......Page 35
Exercises......Page 36
Chapter Notes......Page 39
Pace & Patience......Page 42
Sets......Page 45
Sets, Functions, and Their-Jections......Page 46
Clever Bijections and Counting......Page 54
Proof by Induction and Contradiction......Page 57
Application: Stable Marriages......Page 60
Cultural Review......Page 64
Exercises......Page 65
Chapter Notes......Page 67
Variable Names, Overloading & your Brain......Page 69
TheDefinitionof a Graph......Page 75
Graph Coloring......Page 77
Register Allocation and Hardness......Page 79
Planarity and the Euler Characteristic......Page 81
Application: the Five Color Thorem......Page 83
Approximate Coloring......Page 88
Cultural Review......Page 89
Exercises......Page 90
Chapter Notes......Page 91
Many Subcultures of Mathematics......Page 94
Calculus with 1 Variable......Page 99
Lines and Curves......Page 100
Limits......Page 105
TheDerivative......Page 111
Taylor Series......Page 115
Remainders......Page 120
Application: Finding Roots......Page 122
Exercises......Page 129
Types & Tail Calls......Page 132
Linear Algebra......Page 138
Linear Maps and Vector Spaces......Page 139
Linear Maps, Formally ThisTime......Page 144
TheBasis and Linear Combinations......Page 146
Dimension......Page 150
Matrices......Page 152
Conjugations and Computations......Page 158
One Vector Space to Rule ThemAll......Page 160
Geometry of Vector Spaces......Page 162
Application: Singular Value Decomposition......Page 167
Exercises......Page 182
Chapter Notes......Page 184
Live & learn Linear Algebra......Page 187
Eigenvectors & Eigenvalues......Page 193
Eigenvalues of Graphs......Page 195
Limiting the Scope: Symmetric Matrices......Page 197
Inner Products......Page 200
Orthonormal Bases......Page 204
Computing Eigenvalues......Page 207
TheSpectral Thorem......Page 209
Application: Waves......Page 212
Cultural Review......Page 227
Exercises......Page 228
Chapter Notes......Page 231
Rigor & Formality......Page 233
Generalizing the Derivative......Page 238
Linear Approximations......Page 241
Multivariable Functions and the Chain Rule......Page 246
Computing the Total Derivative......Page 247
TheGeometry of the Gradient......Page 251
Optimizing Multivariable Functions......Page 252
TheChain Rule: a Reprise and a Proof......Page 261
Gradient Descent: an Optimization Hammer......Page 264
Gradients of Computation Graphs......Page 265
Application: Automatic Diffeentiation and a Simple Neural Network......Page 268
Exercises......Page 284
Chapter Notes......Page 287
Argument for Big-O Notation......Page 289
History and Definitio......Page 290
Groups......Page 298
TheGeometric Perspective......Page 300
TheInterface Perspective......Page 304
Homomorphisms: Structure Preserving Functions......Page 306
Building Blocks of Groups......Page 309
Geometry as the Study of Groups......Page 311
TheSymmetry Group of the Poincaré Disk......Page 320
TheHyperbolic Isometry Group as a Group of Matrices......Page 326
Application: Drawing Hyperbolic Tessellations......Page 327
Exercises......Page 343
Chapter Notes......Page 348
New Interface......Page 350
Index......Page 360