Lectures On Linear Algebra

Prominent Russian mathematician's concise, well-written exposition considers: n-dimensional spaces, linear and bilinear forms, linear transformations, canonical form of an arbitrary linear transformation, introduction to tensors, more. Not designed as an introductory text. 1961 edition. I. n-Dimensional Spaces. Linear and Bilinear Forms ? 1. n-Dimensional vector spaces ? 2. Euclidean space ? 3. Orthogonal basis. Isomorphism of Euclidean spaces ? 4. Bilinear and quadratic forms ? 5. Reduction of a quadratic form to a sum of squares ? 6. Reduction of a quadratic form by means of a triangular transformation ? 7. The law of inertia ? 8. Complex n-dimensional space II. Linear Transformations ? 9. Linear transformations. Operations on linear transformations ? 10. Invariant subspaces. Eigenvalues and eigenvectors of a linear transformation ? 11. The adjoint of a linear transformation ? 12. Self-adjoint (Hermitian) transformations. Simultaneous reduction of a pair of quadratic forms to a sum of squares ? 13. Unitary transformations ? 14. Commutative linear transformations. Normal transformations ? 15. Decomposition of a linear transformation into a product of a unitary and self-adjoint transformation ? 16. Linear transformations on a real Euclidean space ? 17. External properties of eigenvalues III. The Canonical Form of an Arbitrary Linear Transformation ? 18. The canonical form of a linear transformation ? 19. Reduction to canonical form ? 20. Elementary divisors ? 21. Polynomial matrices IV. Introduction to Tensors ? 22. The dual space ? 23. Tensors