One A System of Vectors.- 1. Introduction.- 2. Description of the system E3.- 3. Directed line segments and position vectors.- 4. Addition and subtraction of vectors.- 5. Multiplication of a vector by a scalar.- 6. Section formula and collinear points.- 7. Centroids of a triangle and a tetrahedron.- 8. Coordinates and components.- 9. Scalar products.- 10. Postscript.- Exercises on chapter 1.- Two Matrices.- 11. Introduction.- 12. Basic nomenclature for matrices.- 13. Addition and subtraction of matrices.- 14. Multiplication of a matrix by a scalar.- 15. Multiplication of matrices.- 16. Properties and non-properties of matrix multiplication.- 17. Some special matrices and types of matrices.- 18. Transpose of a matrix.- 19. First considerations of matrix inverses.- 20. Properties of nonsingular matrices.- 21. Partitioned matrices.- Exercises on chapter 2.- Three Elementary Row Operations.- 22. Introduction.- 23. Some generalities concerning elementary row operations.- 24. Echelon matrices and reduced echelon matrices.- 25. Elementary matrices.- 26. Major new insights on matrix inverses.- 27. Generalities about systems of linear equations.- 28. Elementary row operations and systems of linear equations.- Exercises on chapter 3.- Four An Introduction to Determinants.- 29. Preface to the chapter.- 30. Minors, cofactors, and larger determinants.- 31. Basic properties of determinants.- 32. The multiplicative property of determinants.- 33. Another method for inverting a nonsingular matrix.- Exercises on chapter 4.- Five Vector Spaces.- 34. Introduction.- 35. The definition of a vector space, and examples.- 36. Elementary consequences of the vector space axioms.- 37. Subspaces.- 38. Spanning sequences.- 39. Linear dependence and independence.- 40. Bases and dimension.- 41. Further theorems about bases and dimension.- 42. Sums of subspaces.- 43. Direct sums of subspaces.- Exercises on chapter 5.- Six Linear Mappings.- 44. Introduction.- 45. Some examples of linear mappings.- 46. Some elementary facts about linear mappings.- 47. New linear mappings from old.- 48. Image space and kernel of a linear mapping.- 49. Rank and nullity.- 50. Row- and column-rank of a matrix.- 50. Row- and column-rank of a matrix.- 52. Rank inequalities.- 53. Vector spaces of linear mappings.- Exercises on chapter 6.- Seven Matrices From Linear Mappings.- 54. Introduction.- 55. The main definition and its immediate consequences.- 56. Matrices of sums, etc. of linear mappings.- 56. Matrices of sums, etc. of linear mappings.- 58. Matrix of a linear mapping w.r.t. different bases.- 58. Matrix of a linear mapping w.r.t. different bases.- 60. Vector space isomorphisms.- Exercises on chapter 7.- Eight Eigenvalues, Eigenvectors and Diagonalization.- 61. Introduction.- 62. Characteristic polynomials.- 62. Characteristic polynomials.- 64. Eigenvalues in the case F = ?.- 65. Diagonalization of linear transformations.- 66. Diagonalization of square matrices.- 67. The hermitian conjugate of a complex matrix.- 68. Eigenvalues of special types of matrices.- Exercises on chapter 8.- Nine Euclidean Spaces.- 69. Introduction.- 70. Some elementary results about euclidean spaces.- 71. Orthonormal sequences and bases.- 72. Length-preserving transformations of a euclidean space.- 73. Orthogonal diagonalization of a real symmetric matrix.- Exercises on chapter 9.- Ten Quadratic Forms.- 74. Introduction.- 75. Change ofbasis and change of variable.- 76. Diagonalization of a quadratic form.- 77. Invariants of a quadratic form.- 78. Orthogonal diagonalization of a real quadratic form.- 79. Positive-definite real quadratic forms.- 80. The leading minors theorem.- Exercises on chapter 10.- Appendix Mappings.- Answers to Exercises.

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