Linear Algebra

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). Language: English