M. A. Akivis and V. V. Goldberg. Translated and Edited by Richard A. Silverman

Editor's Preface
Chapter 1. Linear Spaces
1. Basic Concepts
2. Linear Dependence
3. Dimension and Bases
4. Orthonormal Bases. The Scalar Product
5. The Vector Product. Triple Products
6. Basis Transformations. Tensor Calculus
7. Topics in Analytic Geometry
Chapter 2. Multilinear Forms and Tensors
8. Linear Forms
9. Bilinear Forms
10. Multilinear Forms. General Definition of a Tensor
11. Algebraic Operations on Tensors
12. Symmetric and Antisymmetric Tensors
Chapter 3. Linear Transformations
13. Basic Concepts
14. The Matrix of a Linear Transformation and Its Determinant
15. Linear Transformations and Bilinear Forms
16. Multiplication of Linear Transformations and Matrices
17. Inverse Transformations and Matrices
18. The Group of Linear Transformations and Its Subgroups
Chapter 4. Further Topics
19. Eigenvectors and Eigenvalues
20. The Case of Distinct Eigenvalues
21. Matrix Polynomials and the Hamilton-Cayley Theorem
22. Eigenvectors of a Symmetric Transformation
23. Diagonalization of a Symmetric Transformation
24. Reduction of a Quadratic Form to Canonical Form
25. Representation of a Nonsingular Transformation
Selected Hints and Answers; Bibliography; Index

The present book, a valuable addition to the English-language literature on linear algebra and tensors, constitutes a lucid, eminently readable and completely elementary introduction to this field of mathematics. A special merit of the book is its free use of tensor notation, in particular the Einstein summation convention. The treatment is virtually self-contained. In fact, the mathematical background assumed on the part of the reader hardly exceeds a smattering of calculus and a casual acquaintance with determinants.
The authors begin with linear spaces, starting with basic concepts and ending with topics in analytic geometry. They then treat multilinear forms and tensors (linear and bilinear forms, general definition of a tensor, algebraic operations on tensors, symmetric and antisymmetric tensors, etc.), and linear transformation (again basic concepts, the matrix and multiplication of linear transformations, inverse transformations and matrices, groups and subgroups, etc.). The last chapter deals with further topics in the field: eigenvectors and eigenvalues, matrix ploynomials and the Hamilton-Cayley theorem, reduction of a quadratic form to canonical form, representation of a nonsingular transformation, and more. Each individual section — there are 25 in all — contains a problem set, making a total of over 250 problems, all carefully selected and matched. Hints and answers to most of the problems can be found at the end of the book.
Dr. Silverman has revised the text and numerous pedagogical and mathematical improvements, and restyled the language so that it is even more readable. With its clear exposition, many relevant and interesting problems, ample illustrations, index and bibliography, this book will be useful in the classroom or for self-study as an excellent introduction to the important subjects of linear algebra and tensors.