An Introduction to Linear Algebra

The new edition of this textbook will provide students of varied backgrounds with a comprehensive and accessible introduction to the primary subject- matter of linear algebra. The contents include detailed accounts of matrices and determinants (chapters 2 to 4), finite-dimensional vector spaces (chapter 5), linear mappings (chapters 6 and 7), and (spread through chapters 2, 3, 5 and 6) the applications of these ideas to systems of linear equations, while later chapters (8 to 10) discuss eigenvalues, diagonalization of matrices, euclidean spaces, and quadratic forms. In writing the book, my constant aim has been to draw on my considerable experience of teaching this material to produce an account of it that students striving to gain an understanding of linear algebra will find helpful. In particular, therefore, I have provided suitably detailed explanations of points which students often find difficult. The reader will see that little is taken for granted. For example, the accounts of matrices and vector spaces are self-contained and start right at the begin- ning. However, a basic knowledge of elementary ideas and notations con- cerning sets is assumed, and from chapter 5 onwards the reader must be able to understand simple arguments about sets (such as proving that one set is a subset of another). Then, from chapter 6 onwards, a knowledge of mappings becomes essential, though to help the reader an appendix on mappings pro- vides a condensed summary of everything relevant.

Table of Contents:
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.