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.

Engineers and scientists need to have an introduction to the basics of linear algebra in a context they understand. Computer algebra systems make the manipulation of matrices and the determination of their properties a simple matter, and in practical applications such software is often essential. However, using this tool when learning about matrices, without first gaining a proper understanding of the underlying theory, limits the ability to use matrices and to apply them to new problems. This book explains matrices in the detail required by engineering or science students, and it discusses linear systems of ordinary differential equations. These students require a straightforward introduction to linear algebra illustrated by applications to which they can relate. It caters of the needs of undergraduate engineers in all disciplines, and provides considerable detail where it is likely to be helpful. According to the author the best way to understand the theory of matrices is by working simple exercises designed to emphasize the theory, that at the same time avoid distractions caused by unnecessary numerical calculations. Hence, examples and exercises in this book have been constructed in such a way that wherever calculations are necessary they are straightforward. For example, when a characteristic equation occurs, its roots (the eigenvalues of a matrix) can be found by inspection. The author of this book is Alan Jeffrey, Emeritus Professor of mathematics at the Univesity of Newcastle upon Tyne. He has given courses on engineering mathematics in UK and US Universities.