Introduction to Modern Algebra and Matrix Theory

Description

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. Title Page Copyright Page Editor’s Preface Translators’ Preface Authors’ Preface Table of Contents Chapter I AFFINE SPACE; LINEAR EQUATIONS § 1. n-dimensional Affine Space § 2. Vectors § 3. The Concept of Linear Dependence § 4. Vector Spaces in Rn § 5. Linear Spaces § 6. Linear Equations Homogeneous Linear Equations Non-homogeneous Linear Equations Geometric Applications Chapter II EUCLIDEAN SPACE; THEORY OF DETERMINANTS § 7. Euclidean Length Appendix to § 7: Calculating with the Summation Sign § 8. Volumes and Determinants Fundamental Properties of Determinants Existence and Uniqueness of Determinants Volumes § 9. The Principal Theorems of Determinant Theory The Complete Development of a Determinant The Determinant as a Function of its Column Vectors The Multiplication Theorem The Development of a Determinant by Rows or Columns Determinants and Linear Equations Laplace’s Expansion Theorem § 10. Transformation of Coordinates General Linear Coordinate Systems Cartesian Coordinate Systems Continuous Deformation of a Linear Coordinate System § 11. Construction of Normal Orthogonal Systems and Applications § 12. Rigid Motions Rigid Motions in R2 Rigid Motions in R3 § 13. Affine Transformations Chapter III FIELD THEORY; THE FUNDAMENTAL THEOREM OF ALGEBRA § 14. The Concept of a Field § 15. Polynomials over a Field § 16. The Field of Complex Numbers § 17. The Fundamental Theorem of Algebra Chapter IV ELEMENTS OF GROUP THEORY § 18. The Concept of a Group § 19. Subgroups; Examples § 20. The Basis Theorem for Abelian Groups Chapter V LINEAR TRANSFORMATIONS AND MATRICES § 21. The Algebra of Linear Transformations § 22. Calculation with Matrices Linear Transformations Under a Change of Coordinate System The Determinant of a Linear Transformations Linear Dependence of Matrices Calculation With Matrix Polynomials The Transpose of a Matrix § 23. The Minimal Polynomial; Invariant Subspaces The Minimal Polynomial Invariant Subspaces The Nullspace of a Linear Transformation f(σ) Decomposition of L into Invariant Subspaces Geometric Interpretation § 24. The Diagonal Form and its Applications Unitary Transformations Orthogonal Transformations Hermitian and Symmetric Matrices (Principal Axis Transformations) § 25. The Elementary Divisors of a Polynomial Matrix § 26. The Normal Form Consequences Linear Transformation with Prescribed Elementary Divisors The Jordan Normal Form Index Back Cover