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, and an introduction to tensors. While not designed as an introductory text, the book's well-chosen topics, brevity of presentation, and the author's reputation will recommend it to all students, teachers, and mathematicians working in this sector.
Author(s): Israel M. Gelfand
Series: Interscience tracts in pure and applied mathematics
Edition: 2
Publisher: Interscience Publishers
Year: 1961
Language: English
Pages: 195
City: New York
Tags: Linear Algebra
I. N-dimensional spaces. Linear and bilinear forms......................1
§ 1. N-dimensional vector spaces..........................................1
§ 2. Euclidean space.....................................................14
§ 3. Orthogonal basis. Isomorphism of euclidean spaces...................21
§ 4. Bilinear and quadratic forms........................................34
§ 6. Reduction of a quadratic form to a sum of squares...................42
§ 6. Reduction of a quadratic form by means of a
triangular transformation...........................................46
§ 7. The law of inertia..................................................55
§ 8. Complex n-dimensional space.........................................60
II. Linear transformations..............................................70
§ 9. Linear transformations. Operations on linear transformations........70
§ 10. Invariant subspaces. Eigenvalues and eigenvectors of a
linear transformation...............................................81
§ 11. The adjoint of a linear transformation..............................90
§ 12. Self-adjoint (hermitian) transformations. Simultaneous reduction
of a pair of quadratic forms to a sum of squares....................97
§ 13. Unitary transformations............................................108
§ 14. Commutative linear transformations. Normal transformations.........107
§ 15. Decomposition of a linear transformation into a product of
a unitary and self-adjoint transformation..........................111
§ 16. Linear transformations on a real euclidean space...................114
§ 17. Extremal properties of eigenvalues.................................126
III. The canonical form of an arbitrary linear transformation...........132
§ 18. The canonical form of a linear transformation......................132
§ 19. Reduction to canonical form........................................137
§ 20. Elementary divisors................................................142
§ 21. Polynomial matrices................................................149
IV. Introduction to tensors............................................164
§ 22. The dual space.....................................................164
§ 23. Tensors............................................................171
