Matrix Operations for Engineers and Scientists

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.