1. MATRICES AND LINEAR SYSTEMS.- 1.1 Systems of Algebraic Equations.- 1.2 Suffix and Matrix Notation.- 1.3 Equality, Addition and Scaling of Matrices.- 1.4 Some Special Matrices and the Transpose Operation. Exercises.- 1 2. DETERMINANTS AND LINEAR SYSTEMS.- 2.1 Introduction to Determinants and Systems of Equation.- 2.2 A First Look at Linear Dependence and Independence.- 2.3 Properties of Determinants and the Laplace Expansion Theorem.- 2.4 Gaussian Elimination and Determinants.- 2.5 Homogeneous Systems and a Test for Linear Independence.- 2.6 Determinants and Eigenvalues. Exercises.- 2 3. MATRIX MULTIPLICATION, THE INVERSE MATRIX AND THE NORM.- 3.1 The Inner Product, Orthogonality and the Norm 3.2 Matrix Multiplication.- 3.3 Quadratic Forms.- 3.4 The Inverse Matrix.- 3.5 Orthogonal Matrices 3.6 Matrix Proof of Cramer's Rule.- 3.7 Partitioning of Matrices. Exercises 34. SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS .- 4.1 The Augmented Matrix and Elementary Row Operations.- 4.2 The Echelon and Reduced Echelon Forms of a Matrix.- 4.3 The Row Rank of a Matrix 4.4 Elementary Row Operations and the Inverse Matrix.- 4.5 LU Factorization of a Matrix and its Use When Solving Linear Systems of Algebraic Equations.- 4.6 Eigenvalues and Eigenvectors. Exercises.- 4 5. EIGENVALUES, EIGENVECTORS, DIAGONALIZATION, SIMILARITY AND JORDAN FORMS.- 5.1 Finding Eigenvectors.- 5.2 Diagonalization of Matrices.- 5.3 Quadratic Forms and Diagonalization.- 5.4 The Characteristic Polynomial and the Cayley-Hamilton Theorem.- 5.5 Similar Matrices 5.6 Jordan Normal Forms.- 5.7 Hermitian Matrices. Exercises.-56. SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS.- 6.1 Differentiation and Integration of Matrices.- 6.2 Systems of Homogeneous Constant Coefficient Differential Equations.- 6.3 An Application of Diagonalization 6.4 The Nonhomogeneeous Case.- 6.5 Matrix Methods and the Laplace Transform.- 6.6 The Matrix Exponential and Differential Equations. Exercises.- 6.7. AN INTRODUCTION TO VECTOR SPACES.- 7.1 A Generalization of Vectors.- 7.2 Vector Spaces and a Basis for a Vector Space.- 7.3 Changing Basis Vectors.- 7.4 Row and Column Rank .- .5 The Inner Product.- 7.6 The Angle Between Vectors and Orthogonal Projections.- 7.7 Gram-Schmidt Orthogonalization.- 7.8 Projections.- 7.9 Some Comments on Infinite Dimensional Vector Spaces. Exercises 78. LINEAR TRANSFORMATIONS AND THE GEOMETRY OF THE PLANE.- 8.1 Rotation of Coordinate Axes.- 8.2 The Linearity of the Projection Operation.- 8.3 Linear Transformations 8.4 Linear Transformations and the Geometry of the Plane. Exercises.- 8Solutions to all Exercises.
About the Author: Alan Jeffrey is emeritus professor of engineering mathematics at Newcastle. He has had a distinguished career which included work at University of Delaware, Stanford, Wisconsin and City University Hong Kong. He has published 14 books, some at undergraduate level and others at the research monograph level, some with Springer, and his sales records look very good. He also is well known because he edited some important reference works such as the Handbook of Mathematical Formula.