Linear Algebra

This textbook gives a detailed and comprehensive presentation of linear algebra based on an axiomatic treatment of linear spaces. For this fourth edition some new material has been added to the text, for instance, the intrinsic treatment of the classical adjoint of a linear transformation in Chapter IV, as well as the discussion of quaternions and the classifica­ tion of associative division algebras in Chapter VII. Chapters XII and XIII have been substantially rewritten for the sake of clarity, but the contents remain basically the same as before. Finally, a number of problems covering new topics-e.g. complex structures, Caylay numbers and symplectic spaces - have been added. I should like to thank Mr. M. L. Johnson who made many useful suggestions for the problems in the third edition. I am also grateful to my colleague S. Halperin who assisted in the revision of Chapters XII and XIII and to Mr. F. Gomez who helped to prepare the subject index. Finally, I have to express my deep gratitude to my colleague J. R. Van­ stone who worked closely with me in the preparation of all the revisions and additions and who generously helped with the proof reading.

0. Prerequisites.- I. Vector spaces.- § 1. Vector spaces.- § 2. Linear mappings.- § 3. Subspaces and factor spaces.- § 4. Dimension.- § 5. The topology of a real finite dimensional vector space.- II. Linear mappings.- § 1. Basic properties.- § 2. Operations with linear mappings.- § 3. Linear isomorphisms.- § 4. Direct sum of vector spaces.- § 5. Dual vector spaces.- § 6. Finite dimensional vector spaces.- III. Matrices.- § 1. Matrices and systems of linear equations.- § 2. Multiplication of matrices.- § 3. Basis transformation.- § 4. Elementary transformations.- IV. Determinants.- § 1. Determinant functions.- § 2. The determinant of a linear transformation.- § 3. The determinant of a matrix.- § 4. Dual determinant functions.- § 5. The adjoint matrix.- § 6. The characteristic polynomial.- § 7. The trace.- § 8. Oriented vector spaces.- V. Algebras.- § 1. Basic properties.- § 2. Ideals.- § 3. Change of coefficient field of a vector space.- VI. Gradations and homology.- § 1. G-graded vector spaces.- § 2. G-graded algebras.- § 3. Differential spaces and differential algebras.- VII. Inner product spaces.- § 1. The inner product.- § 2. Orthonormal bases.- § 3. Normed determinant functions.- § 4. Duality in an inner product space.- § 5. Normed vector spaces.- § 6. The algebra of quaternions.- VIII. Linear mappings of inner product spaces.- § 1. The adjoint mapping.- § 2. Selfadjoint mappings.- § 3. Orthogonal projections.- § 4. Skew mappings.- § 5. Isometric mappings.- § 6. Rotations of Euclidean spaces of dimension 2, 3 and 4.- § 7. Differentiate families of linear automorphisms.- IX. Symmetric bilinear functions.- § 1. Bilinear and quadratic functions.- § 2. The decomposition of E.- § 3. Pairs of symmetric bilinear functions.- §4. Pseudo-Euclidean spaces.- § 5. Linear mappings of Pseudo-Euclidean spaces.- X. Quadrics.- § 1. Affine spaces.- § 2. Quadrics in the affine space.- § 3. Affine equivalence of quadrics.- § 4. Quadrics in the Euclidean space.- XI. Unitary spaces.- § 1. Hermitian functions.- § 2. Unitary spaces.- § 3. Linear mappings of unitary spaces.- § 4. Unitary mappings of the complex plane.- § 5. Application to Lorentz-transformations.- XII. Polynomial algebra.- § 1. Basic properties.- § 2. Ideals and divisibility.- § 3. Factor algebras.- § 4. The structure of factor algebras.- XIII. Theory of a linear transformation.- § 1. Polynomials in a linear transformation.- § 2. Generalized eigenspaces.- § 3. Cyclic spaces.- § 4. Irreducible spaces.- § 5. Application of cyclic spaces.- § 6. Nilpotent and semisimple transformations.- § 7. Applications to inner product spaces.