Linear Algebra

David Poole is Professor Emeritus of Mathematics at Trent University, where he was a faculty member from 1984–2016. Prof. 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. For his services to mathematics education, Prof. Poole was named a Fellow of the Canadian Mathematical Society in 2018. Prof. Poole served as chair of Trent’s Mathematics Department on three occasions, and from 2002–2007 he was Associate Dean of Arts and Science (Teaching & Learning). His dedication to high quality teaching extends to policy development: he has been involved with teaching committees, task forces on mathematics education, and he has produced reports on teaching and learning approaches. His research interests include discrete mathematics, ring theory, and mathematics education. He received his B.Sc. from Acadia University (1976) before earning his M.Sc. (1977) and Ph.D. (1984) from McMaster University. When he is not doing mathematics, David Poole’s hobbies include cooking and constructing crossword puzzles, 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. Writing Project: Origins of the Dot Product and the Cross Product. Applications.2. SYSTEMS OF LINEAR EQUATIONS.Introduction: Triviality. Introduction to Systems of Linear Equations. Direct Methods for Solving Linear Systems. Writing Project: A History of Gaussian Elimination. Explorations: Lies My Computer Told Me; Partial Pivoting; Counting Operations: An Introduction to the Analysis of Algorithms. Spanning Sets and Linear Independence. Applications. 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.4. EIGENVALUES AND EIGENVECTORS.Introduction: A Dynamical System on Graphs. Introduction to Eigenvalues and Eigenvectors. Determinants. Writing Project: Which Came First-the Matrix or the Determinant? Vignette: Lewis Carroll's Condensation Method. Exploration: Geometric Applications of Determinants. Eigenvalues and Eigenvectors of n x n Matrices. Writing Project: The History of Eigenvalues. Similarity and Diagonalization. Iterative Methods for Computing Eigenvalues. Applications and the Perron-Frobenius Theorem.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. Explorations: The Modified QR Factorization; Approximating Eigenvalues with the QR Algorithm. Orthogonal Diagonalization of Symmetric Matrices. Applications.6. VECTOR SPACES.Introduction: Fibonacci in (Vector) Space. Vector Spaces and Subspaces. Linear Independence, Basis, and Dimension. Writing Project: The Rise of Vector Spaces. 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.7. DISTANCE AND APPROXIMATION.Introduction: Taxicab Geometry. Inner Product Spaces. Explorations: Vectors and Matrices with Complex Entries; Geometric Inequalities and Optimization Problems. Norms and Distance Functions. Least Squares Approximation. The Singular Value Decomposition. Vignette: Digital Image Compression. Applications.8. CODES. (Online)Code Vectors. Vignette: The Codabar System. Error-Correcting Codes. Dual Codes. Linear Codes. The Minimum Distance of a Code.Appendix A: Mathematical Notation and Methods of Proof.Appendix B: Mathematical Induction.Appendix C: Complex Numbers.Appendix D: Polynomials.