Edward Barry Saff, PhD, is Professor of Mathematics and Director of the Center for Constructive Approximation at Vanderbilt University. Dr. Saff is an Inaugural Fellow of the American Mathematical Society, Foreign Member of the Bulgarian Academy of Science, and the recipient of both a Guggenheim and Fulbright Fellowship. He is Editor-in-Chief of two research journals, Constructive Approximation and Computational Methods and Function Theory, and has authored or coauthored over 250 journal articles and eight books. Dr. Saff also serves as an organizer for a sequence of international research conferences that help to foster the careers of mathematicians from developing countries.Arthur Da vid Snider, PhD, PE, is Professor Emeritus at the University of South Florida, where he served on the faculties of the Departments of Mathematics, Physics, and Electrical Engineering. Previously an analyst at the Massachusetts Institute of Technology’s Draper Lab and recipient of the USF Krivanek Distinguished Teacher Award, he consults in industry and has authored or coauthored over 100 journal articles and eight books. With the support of the National Science Foundation, Dr. Snider also pioneered a course in fine art appreciation for engineers.

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• 1 Systems of Linear Algebraic Equations 11.1 Linear Algebraic Equations 11.2 Matrix Representation of Linear Systems and the Gauss-Jordan Algorithm 121.3 The Complete Gauss Elimination Algorithm 171.4 Echelon Form and Rank 261.5 Computational Considerations 342 Matrix Algebra 392.1 Matrix Multiplication 392.2 Some Useful Applications of Matrix Operators 482.3 The Inverse and the Transpose 542.4 Determinants 612.5 Three Important Determinant Rules 68PART I REVIEW PROBLEMS FOR PART I 793 Vector Spaces 893.1 General Spaces, Subspaces, and Spans 893.2 Linear Dependence 933.3 Bases, Dimension, and Rank 974 Orthogonality 1054.1 Orthogonal Vectors and the Gram-Schmidt Algorithm 1054.2 Orthogonal Matrices 1154.3 Least Squares 1234.4 Function Spaces 133PART II REVIEW PROBLEMS FOR PART II 1405 Eigenvectors and Eigenvalues 1445.1 Eigenvector Basics 1445.2 Calculating Eigenvalues and Eigenvectors 1555.3 Symmetric and Hermitian Matrices 1686 Similarity 1816.1 Similarity Transformations and Diagonalizability 1816.2 Principal Axes and Normal Modes 1896.3 Schur Decomposition and Its Implications 1986.4 The Singular Value Decomposition 2126.5 The Power Method and the QR Algorithm 2177 Linear Systems of Differential Equations 2217.1 First-Order Linear Systems 2217.2 The Matrix Exponential Function 2307.3 The Jordan Normal Form 2367.4 Matrix Exponentiation via Generalized Eigenvectors 246PART III REVIEW PROBLEMS FOR PART III 255