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.
Author(s): Otto Schreier, Emanuel Sperner
Series: Dover Books on Mathematics Series
Edition: 2
Publisher: Dover
Year: 2011
Language: English
Pages: 539
City: Mineola, New York
Tags: Linear Algebra, Matrix Theory, Algebra
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
