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
 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. SkewSymmetric Tensors and Symmetric Tensors
 Section 37. The SkewSymmetric 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
PART II VECTOR AND TENSOR ALGEBRA
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Introduction to Vectors and Tensors, Vol 1 Linear and Multilinear Algebra