Abstract Algebra and Discrete Mathematics- Mathematical Structures Chapter 1 of Abstract Algebra and Discrete Mathematics

This book is chapter 1 of a series of book. The entire series is an introduction to combinatory, number theory, topology, groups, rings, fields, modules, algebraic and integral extensions, noncommutative algebra, algebraic number theory, algebraic geometry, algebraic topology, and even more. The title of the series is Abstract Algebra and Discrete Mathematics.The focus is on breadth, rather than depth. Excellent books already exist for any one of these topics in detail, and I don't want to reinvent that wheel. Instead, this book knits them all together, providing a foundation for each topic in turn. By analogy, you might point your backyard telescope to every corner of the galaxy, in an effort to comprehend its scope, beauty, and diversity. You might not understand the Crab Nebula in all its detail, this is not the Hubble Space Telescope, but you should walk away with an appreciation for the vastness and the wonder of the galaxy, in this case, the galaxy of modern mathematics. If, from time to time, you find yourself saying, ''How did anybody ever think of that?'', then I have succeeded.Each chapter builds on the information that has gone before, and forward references are rare, though they do happen from time to time. I hope this series of books is more accessible than a sea of disconnected web pages, which is the hallmark of most math websites.This book is Chapter 1, Data Structures,By Karl Dahlke and Kermit Rose, Copyright © 2015Table of ContentsPage 04 1.01 Properties of ArithmeticPage 05 1.02 Modular MathPage 05 1.03 HomomorphismPage 08 1.04 Casting out NinesPage 09 1.05 Monomorphism, Epimorphism, IsomorphismPage 11 1.06 Discrete LogsPage 13 1.07 Binary SearchPage 14 1.08 Function CompositionPage 14 1.09 PermutationsPage 17 1.10 Even and Odd PermutationsPage 19 1.11 MatricesPage 23 1.12 Linear FunctionsPage 25 1.13 Matrix as FunctionPage 27 1.14 DeterminantPage 33 1.15 Gaussian EliminationPage 36 1.16 Elementary Row OperationsPage 37 1.17 The Determinant of the ProductPage 38 1.18 Matrix Identities and Inverses Page 43 1.19 OrthogonalPage 44 1.20 Determinant equals VolumePage 46 1.21 The Gram Schmidt ProcessPage 47 1.22 The Shoelace FormulaPage 50 1.23 Orthonormal and Rotations Page 55 1.24 Cross Product Page 57 1.25 Vandermonde MatrixPage 58 1.26 A Matrix of Matrices Page 59 1.27 Complex Numbers and the Gaussian IntegersPage 61 1.28 Eisenstein IntegersPage 63 1.29 QuaternionsPage 64 1.30 Half Integer QuaternionsPage 65 1.31 Projective Space and an Interesting HomomorphismPage 68 1.32 The Square Root of a Rotation Page 69 1.33 The Hairy Ball Theorem Page 70 1.34 Generalized Euclidean SpacePage 71 1.35 PolynomialsPage 72 1.36 Synthetic Division, Roots, and GCDPage 74 1.37 Formal DerivativePage 75 1.38 Power and Laurent SeriesPage 76 1.39 p-adic Numbers