# Book:M.A. Akivis/An Introduction to Linear Algebra & Tensors

## M.A. Akivis and V.V. Goldberg: An Introduction to Linear Algebra & Tensors

Published $\text {1972}$, Dover Publications

ISBN 0-486-63545-7 (translated by Richard A. Silverman)

Translated from:

Tensor Calculus

### Contents

Editor's Preface (Richard A. Silverman)
$1$ 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
$2$ 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 Asymmetric Tensors
$3$ 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
$4$ 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