Introduction to Modern Algebra and Matrix Theory Second Edition

by Schreier, O.; Sperner, E.; David, Martin; Hausner, Melvin

• ISBN13: 9780486482200
• ISBN10: 0486482200
• Edition: 2nd
• Format: Paperback
• Copyright: 2011-07-19
• Publisher: Dover Publications

Summary

This unique text provides students with a basic course in both calculus and analytic geometry -- no competitive editions cover both topics in a single volume. Its prerequisites are minimal, and the order of its presentation promotes an intuitive approach to calculus. Algebraic concepts receive an unusually strong emphasis. Numerous exercises appear throughout the text. 1951 edition.

Author Biography

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	p. iii
Translators' Preface	p. iii
Authors' Preface	p. v
Affine Space; Linear Equations
n-dimensional Affine Space	p. 1
Vectors	p. 6
The Concept of Linear Dependence	p. 16
Vector Spaces in Rn	p. 19
Linear Spaces	p. 27
Linear Equations	p. 34
Homogeneous Linear Equations	p. 36
Non-homogeneous Linear Equations	p. 40
Geometric Applications	p. 44
Euclidean Space; Theory of Determinants
Euclidean Length	p. 50
Calculating with the Summation Sign	p. 60
Volumes and Determinants	p. 63
Fundamental Properties of Determinants	p. 69
Existence and Uniqueness of Determinants	p. 74
Volumes	p. 83
The Principal Theorems of Determinant Theory	p. 87
The Complete Development of a Determinant	p. 87
The Determinant as a Function of its Column Vectors	p. 89
The Multiplication Theorem	p. 96
The Development of a Determinant by Rows or Columns	p. 98
Determinants and Linear Equations	p. 100
Laplace's Expansion Theorem	p. 105
Transformation of Coordinates	p. 117
General Linear Coordinate Systems	p. 117
Cartesian Coordinate Systems	p. 126
Continuous Deformation of a Linear Coordinate System	p. 131
Construction of Normal Orthogonal Systems and Applications	p. 140
Rigid Motions	p. 153
Rigid Motions in R2	p. 162
Rigid Motions in R3	p. 168
Affine Transformations	p. 180
Field Theory; The Fundamental Theorem of Algebra
The Concept of a Field	p. 187
Polynomials over a Field	p. 204
The Field of Complex Numbers	p. 218
The Fundamental Theorem of Algebra	p. 230
Elements of Group Theory
The Concept of a Group	p. 245
Subgroups; Examples	p. 251
The Basis Theorem for Abelian Groups	p. 260
Linear Transformations and Matrices
The Algebra of Linear Transformations	p. 273
Calculation with Matrices	p. 283
Linear Transformations Under a Change of Coordinate System	p. 293
The Determinant of a Linear Transformations	p. 296
Linear Dependence of Matrices	p. 297
Calculation With Matrix Polynomials	p. 298
The Transpose of a Matrix	p. 301
The Minimal Polynomial; Invariant Subspaces	p. 303
The Minimal Polynomial	p. 303
Invariant Subspaces	p. 305
The Nullspace of a Linear Transformation f(¿)	p. 306
Decomposition of L into Invariant Subspaces	p. 310
Geometric Interpretation	p. 315
The Diagonal Form and its Applications	p. 320
Unitary Transformations	p. 327
Orthogonal Transformations	p. 334
Hermitian and Symmetric Matrices (Principal Axis Transformations)	p. 340
The Elementary Divisors of a Polynomial Matrix	p. 344
The Normal Form	p. 355
Consequences	p. 363
Linear Transformation with Prescribed Elementary Divisors	p. 365
The Jordan Normal Form	p. 367
Index	p. 373
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