David Poole's innovative book prepares students to make the transition from the computational aspects of the course to the theoretical by emphasizing vectors and geometric intuition from the start. Designed for a one- or two-semester introductory course and written in simple, "mathematical English" the book presents interesting examples before abstraction. This immediately follows up theoretical discussion with further examples and a variety of applications drawn from a number of disciplines, which reinforces the practical utility of the math, and helps students from a variety of backgrounds and learning styles stay connected to the concepts they are learning. Poole's approach helps students succeed in this course by learning vectors and vector geometry first in order to visualize and understand the meaning of the calculations that they will encounter and develop mathematical maturity for thinking abstractly.
Designed for either a one- or two-semester course, this college-level introductory algebra textbook covers vectors, linear equations, matrices, eigenvalues and eigenvectors, orthogonality, vector spaces, and distance and approximation. Appendices discuss mathematical notation and methods of proof, mathematical induction, complex numbers, polynomials, and technology. Concrete examples are emphasized over abstract principles or theoretical concerns. Poole teaches at Trent University. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Product dimensions: 8.20 (w) x 10.10 (h) x 1.40 (d)
Meet the Author
David Poole is Professor of Mathematics at Trent University, where he has been a faculty member since 1984. Dr. Poole has won numerous teaching awards: Trent University's Symons Award for Excellence in Teaching (the university's top teaching award), three merit awards for teaching excellence, a 2002 Ontario Confederation of University Faculty Associations Teaching Award (the top university teaching award in the province), a 2003 3M Teaching Fellowship (the top university teaching award in Canada, sponsored by 3M Canada Ltd.), a 2007 Leadership in Faculty Teaching Award from the province of Ontario, and the Canadian Mathematical Society's 2009 Excellence in Teaching Award. From 2002-2007, Dr. Poole was Trent University's Associate Dean (Teaching & Learning). His research interests include discrete mathematics, ring theory, and mathematics education. He received his B.Sc. from Acadia University in 1976 before earning his M.Sc. (1977) and Ph.D. (1984) from McMaster University. When he is not doing mathematics, David Poole enjoys hiking and cooking, and he is an avid film buff.
1. VECTORS. Introduction: The Racetrack Game. The Geometry and Algebra of Vectors. Length and Angle: The Dot Product. Exploration: Vectors and Geometry. Lines and Planes. Exploration: The Cross Product. Applications: Force Vectors; Code Vectors. Vignette: The Codabar System. 2. SYSTEMS OF LINEAR EQUATIONS. Introduction: Triviality. Introduction to Systems of Linear Equations. Direct Methods for Solving Linear Systems. Exploration: Lies My Computer Told Me. Exploration: Partial Pivoting. Exploration: Counting Operations: An Introduction to the Analysis of Algorithms. Spanning Sets and Linear Independence. Applications: Allocation of Resources; Balancing Chemical Equations; Network Analysis; Electrical Networks; Linear Economic Models; Finite Linear Games. Vignette: The Global Positioning System. Iterative Methods for Solving Linear Systems. 3. MATRICES. Introduction: Matrices in Action. Matrix Operations. Matrix Algebra. The Inverse of a Matrix. The LU Factorization. Subspaces, Basis, Dimension, and Rank. Introduction to Linear Transformations. Vignette: Robotics. Applications: Markov Chains; Linear Economic Models; Population Growth; Graphs and Digraphs; Error-Correcting Codes. 4. EIGENVALUES AND EIGENVECTORS. Introduction: A Dynamical System on Graphs. Introduction to Eigenvalues and Eigenvectors. Determinants. Vignette: Lewis Carroll's Condensation Method. Exploration: Geometric Applications of Determinants. Eigenvalues and Eigenvectors of n x n Matrices. Similarity and Diagonalization. Iterative Methods for Computing Eigenvalues. Applications and the Perron-Frobenius Theorem: Markov Chains; Population Growth; The Perron-Frobenius Theorem; Linear Recurrence Relations; Systems of Linear Differential Equations; Discrete Linear Dynamical Systems. Vignette: Ranking Sports Teams and Searching the Internet. 5. ORTHOGONALITY. Introduction: Shadows on a Wall. Orthogonality in Rn. Orthogonal Complements and Orthogonal Projections. The Gram-Schmidt Process and the QR Factorization. Exploration: The Modified QR Factorization. Exploration: Approximating Eigenvalues with the QR Algorithm. Orthogonal Diagonalization of Symmetric Matrices. Applications: Dual Codes; Quadratic Forms; Graphing Quadratic Equations. 6. VECTOR SPACES. Introduction: Fibonacci in (Vector) Space. Vector Spaces and Subspaces. Linear Independence, Basis, and Dimension. Exploration: Magic Squares. Change of Basis. Linear Transformations. The Kernel and Range of a Linear Transformation. The Matrix of a Linear Transformation. Exploration: Tilings, Lattices and the Crystallographic Restriction. Applications: Homogeneous Linear Differential Equations; Linear Codes. 7. DISTANCE AND APPROXIMATION. Introduction: Taxicab Geometry. Inner Product Spaces. Exploration: Vectors and Matrices with Complex Entries. Exploration: Geometric Inequalities and Optimization Problems. Norms and Distance Functions. Least Squares Approximation. The Singular Value Decomposition. Vignette: Digital Image Compression. Applications: Approximation of Functions; Error-Correcting Codes. Appendix A: Mathematical Notation and Methods of Proof. Appendix B: Mathematical Induction. Appendix C: Complex Numbers. Appendix D: Polynomials.