Introduction to Modern Algebra and Matrix Theory: Second Edition

by O. Schreier, E. Sperner, Martin David (Translator), Melvin Hausner (Translator)

Overview

This unique text provides students with a single-volume treatment of the basics of calculus and analytic geometry. It reflects the teaching methods and philosophy of Otto Schreier, an influential mathematician and professor. The order of its presentation promotes an intuitive approach to calculus, and it offers a strong emphasis on algebra with minimal prerequisites.
Starting with affine space and linear equations, the text proceeds to considerations of Euclidean space and the theory of determinants, field theory and the fundamental theorem of algebra, elements of group theory, and linear transformations and matrices. Numerous exercises at the end of each section form important supplements to the text.

Product Details

ISBN-13:	9780486482200
Publisher:	Dover Publications
Publication date:	07/19/2011
Series:	Dover Books on Mathematics Series
Edition description:	Second Edition
Pages:	398
Product dimensions:	6.00(w) x 9.00(h) x 0.80(d)

About the Author

German mathematician Otto Schreier (1901-29) made important contributions to combinatorial group theory in the course of his tragically short life. This book was assembled from his lecture notes by his student, Emmanuel Sperner.

Table of Contents

Editor's Preface iii

Translators' Preface iii

Authors' Preface v

Chapter I Affine Space; Linear Equations

§ 1 n-dimensional Affine Space 1

§ 2 Vectors 6

§ 3 The Concept of Linear Dependence 16

§ 4 Vector Spaces in R_n 19

§ 5 Linear Spaces 27

§ 6 Linear Equations 34

Homogeneous Linear Equations 36

Non-homogeneous Linear Equations 40

Geometric Applications 44

Chapter II Euclidean Space; Theory of Determinants

§ 7 Euclidean Length 50

Appendix to § 7 Calculating with the Summation Sign 60

§ 8 Volumes and Determinants 63

Fundamental Properties of Determinants 69

Existence and Uniqueness of Determinants 74

Volumes 83

§ 9 The Principal Theorems of Determinant Theory 87

The Complete Development of a Determinant 87

The Determinant as a Function of its Column Vectors 89

The Multiplication Theorem 96

The Development of a Determinant by Rows or Columns 98

Determinants and Linear Equations 100

Laplace's Expansion Theorem 105

§ 10 Transformation of Coordinates 117

General Linear Coordinate Systems 117

Cartesian Coordinate Systems 126

Continuous Deformation of a Linear Coordinate System 131

§ 11 Construction of Normal Orthogonal Systems and Applications 140

§ 12 Rigid Motions 153

Rigid Motions in R₂ 162

Rigid Motions in R₃ 168

§ 13 Affine Transformations 180

Chapter III Field Theory; The Fundamental Theorem of Algebra

§ 14 The Concept of a Field 187

§ 15 Polynomials over a Field 204

§ 16 The Field of Complex Numbers 218

§ 17 The Fundamental Theorem of Algebra 230

Chapter IV Elements of Group Theory

§ 18 The Concept of a Group 245

§ 19 Subgroups; Examples 251

§ 20 The Basis Theorem for Abelian Groups 260

Chapter V Linear Transformations and Matrices

§ 21 The Algebra of Linear Transformations 273

§ 22 Calculation with Matrices 283

Linear Transformations Under a Change of Coordinate System 293

The Determinant of a Linear Transformations 296

Linear Dependence of Matrices 297

Calculation With Matrix Polynomials 298

The Transpose of a Matrix 301

§ 23 The Minimal Polynomial; Invariant Subspaces 303

The Minimal Polynomial 303

Invariant Subspaces 305

The Nullspace of a Linear Transformation f(σ) 306

Decomposition of L into Invariant Subspaces 310

Geometric Interpretation 315

§ 24 The Diagonal Form and its Applications 320

Unitary Transformations 327

Orthogonal Transformations 334

Hermitian and Symmetric Matrices (Principal Axis Transformations) 340

§ 25 The Elementary Divisors of a Polynomial Matrix 344

§ 26 The Normal Form 355

Consequences 363

Linear Transformation with Prescribed Elementary Divisors 365

The Jordan Normal Form 367

Index 373