Lectures on Linear Algebra

Title page......Page 1
Preface to the second edition......Page 3
Preface to the first edition......Page 5
Contents......Page 7
§ 1. n-Dimensional vector spaces......Page 9
§ 2. Euclidean space......Page 22
§ 3. Orthogonal basis. Isomorphism of Euclidean spaces......Page 29
§ 4. Bilinear and quadratic forms......Page 42
§ 5. Reduction of a quadratic form to a sum of squares......Page 50
§ 6. Reduction of a quadratic form by means of a triangular transformation......Page 54
§ 9. The law of inertia......Page 63
§ 8. Complex ,dimensional space......Page 68
§ 9. Linear transformations. Operations on linear transformations......Page 78
§ 10. Invariant subspaces. Eigenvalues and eigenvectors of a linear transformation......Page 89
§ 11. The adjoint of a linear transformation......Page 98
§ 12. Self-adjoint (Hermitian) transformations. Simultaneous reduction of a pair of quadratic forms to a sum of squares......Page 105
§ 13. Unitary transformations......Page 111
§ 14. Commutative linear transformations. Normal transformations......Page 115
§ 15. Decomposition of a linear transformation into a product of a unitary and self-adjoint transformation......Page 119
§ 16. Linear transformations on a real Euclidean space......Page 122
§ 17. Extremal properties of eigenvalues......Page 134
§ 18. The canonical form of a linear transformation......Page 140
§ 19. Reduction to canonical form......Page 145
§ 20. Elementary divisors......Page 150
§ 21. Polynomial matrices......Page 157
§ 22. The dual space......Page 172
§ 23. Tensors......Page 179