Introduction to Linear Algebra and Tensors

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