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

Written by: Ray M. Bowen (Texas A&M University) and C.C. Wang (Rice University)

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.”

### 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

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