Introduction to Vectors and Tensors, Vol 1 Linear and Multilinear Algebra

One of several engineering and advanced algebra textbooks written and generously shared by Ray M. Bowen of Texas A&M University. Bowen was joined by fellow educator C.C. Wang in the writing of this textbook. Introduction to Vectors and Tensors, Vol 1 is the first of two volumes which discuss the basic concepts of vector and tensor analysis. The second volume Introduction to Vectors and Tensors, Vol 2. Vector and Tensor Analysis is also featured here on The Free Online Textbook List.

The goal of this textbook is to provide Engineering, Physical Science and Social Science students with “a modern introduction to vectors and tensors” with a focus on their applications. Volume 1 includes information often found in an introductory linear algebra course. Volume II covers vector and tensor analysis.

Bowen, a member of the mechanical engineering faculty, wrote this text to focus on the application of the topics rather than offering “a collection of mathematical manipulations of long equations decorated by a multitude of subscripts and superscripts.”

Table of Contents Introduction to Vectors and Tensors, Vol.1

PART I BASIC MATHEMATICS

CHAPTER 0 Elementary Matrix Theory

CHAPTER 1 Sets, Relations, and Functions

Section 1. Sets and Set Algebra

Section 2. Ordered Pairs” Cartesian Products” and Relations

Section 3. Functions

CHAPTER 2 Groups, Rings and Fields

Section 4. The Axioms for a Group

Section 5. Properties of a Group

Section 6. Group Homomorphisms

Section 7. Rings and Fields

PART II VECTOR AND TENSOR ALGEBRA

CHAPTER 3 Vector Spaces

Section 8. The Axioms for a Vector Space

Section 9. Linear Independence, Dimension and Basis

Section 10. Intersection, Sum and Direct Sum of Subspaces

Section 11. Factor Spaces

Section 12. Inner Product Spaces

Section 13. Orthogonal Bases and Orthogonal Compliments

Section 14. Reciprocal Basis and Change of Basis

CHAPTER 4. Linear Transformations

Section 15. Definition of a Linear Transformation

Section 16. Sums and Products of Linear Transformations

Section 17. Special Types of Linear Transformations

Section 18. The Adjoint of a Linear Transformation

Section 19. Component Formulas

CHAPTER 5. Determinants and Matrices

Section 20.The Generalized Kronecker Deltas and the Summation Convention

Section 21. Determinants

Section 22. The Matrix of a Linear Transformation

Section 23. Solution of Systems of Linear Equations

CHAPTER 6 Spectral Decompositions

Section 24. Direct Sum of Endomorphisms

Section 25. Eigenvectors and Eigenvalues

Section 26. The Characteristic Polynomial

Section 27. Spectral Decomposition for Hermitian Endomorphisms

Section 28. Illustrative Examples

Section 29. The Minimal Polynomial

Section 30. Spectral Decomposition for Arbitrary Endomorphisms

CHAPTER 7 Tensor Algebra

Section 31. Linear Functions, the Dual Space

Section 32. The Second Dual Space, Canonical Isomorphisms

Section 33. Multilinear Functions, Tensors

Section 34. Contractions

Section 35. Tensors on Inner Product Spaces

CHAPTER 8. Exterior Algebra

Section 36. Skew-Symmetric Tensors and Symmetric Tensors

Section 37. The Skew-Symmetric Operator

Section 38. The Wedge Product

Section 39. Product Bases and Strict Components

Section 40. Determinants and Orientations

Section 41. Duality

Section 42. Transformation to Contravariant Representation

View this Free Online Material at the source:
 
Introduction to Vectors and Tensors, Vol 1 Linear and Multilinear Algebra