Linear Algebra

Linear algebra is the branch of mathematics that has grown from a care ful study of the problem of solving systems of linear equations. The ideas that developed in this way have become part of the language of much of higher mathematics. They also provide a framework for appli cations of linear algebra to many problems in mathematics, the natural sciences, economics, and computer science. This book is the revised fourth edition of a textbook designed for upper division courses in linear algebra. While it does not presuppose an earlier course, many connections between linear algebra and under graduate analysis are worked into the discussion, making it best suited for students who have completed the calculus sequence. For many students, this may be the first course in which proofs of the main results are presented on an equal footing with methods for solving numerical problems. The concepts needed to understand the proofs are shown to emerge naturally from attempts to solve concrete problems. This connection is illustrated by worked examples in almost every section. Many numerical exercises are included, which use all the ideas, and develop important techniques for problem-solving. There are also theoretical exercises, which provide opportunities for students to discover interesting things for themselves, and to write mathematical explanations in a convincing way. Answers and hints for many of the problems are given in the back. Not all answers are given, however, to encourage students to learn how to check their work.

1. Introduction to Linear Algebra. - 1. Some problems which lead to linear algebra. - 2. Number systems and mathematical induction. - 2. Vector Spaces and Systems of Linear Equations. - 3. Vector spaces. - 4. Subspaces and linear dependence. - 5. The concepts of basis and dimension. - 6. Row equivalence of matrices. - 7. Some general theorems about finitely generated vector spaces. - 8. Systems of linear equations. - 9. Systems of homogeneous equations. - 10. Linear manifolds. - 3. Linear Transformations and Matrices. - 11. Linear transformations. - 12. Addition and multiplication of matrices. - 13. Linear transformations and matrices. - 4. Vector Spaces with an Inner Product. - 14. The concept of symmetry. - 15. Inner products. - 5. Determinants. - 16. Definition of determinants. - 17. Existence and uniqueness of determinants. - 18. The multiplication theorem for determinants. - 19. Further properties of determinants. - 6. Polynomials and Complex Numbers. - 20. Polynomials. - 21. Complex numbers. - 7. The Theory of a Single Linear Transformation. - 22. Basic concepts. - 23. Invariant subspaces. - 24. The triangular form theorem. - 25. The rational and Jordan canonical forms. - 8. Dual Vector Spaces and Multilinear Algebra. - 26. Quotient spaces and dual vector spaces. - 27. Bilinear forms and duality. - 28. Direct sums and tensor products. - 29. A proof of the elementary divisor theorem. - 9. Orthogonal and Unitary Transformations. - 30. The structure of orthogonal transformations. - 31. The principal axis theorem. - 32. Unitary transformations and the spectral theorem. - 10. Some Applications of Linear Algebra. - 33. Finite symmetry groups in three dimensions. - 34. Application to differential equations. - 35. Analytic methods in matrix theory. - 36. Sums of squares and Hurwitz s theorem. - Bibliography (with Notes). - Solutions of Selected Exercises. - Symbols (Including Greek Letters).