Linear Algebra: A Modern Introduction

David Poole's innovative book emphasizes vectors and geometric intuition from the start and better prepares students to make the transition from the computational aspects of the course to the theoretical. Poole covers vectors and vector geometry first to enable students to visualize the mathematics while they are doing matrix operations. With a concrete understanding of vector geometry, students are able to visualize and understand the meaning of the calculations that they will encounter. By seeing the mathematics and understanding the underlying geometry, students develop mathematical maturity and can think abstractly when they reach vector spaces. Throughout the text, Poole's direct conversational writing style connects with students, and an abundant selection of applications from a broad range of disciplines clearly demonstrates the relevance of linear algebra.

Table of Contents:
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. Code Vectors and Modular Arithmetic. Vignette: The Codabar System. Chapter Review. 2. SYSTEMS OF LINEAR EQUATIONS. Introduction: Triviality. Introduction to Systems of Linear Equations. Exploration: Lies My Computer Told Me. Direct Methods for Solving Linear Systems. 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; Finite Linear Games. Vignette: The Global Positioning System. Iterative Methods for Solving Linear Systems. Chapter Review. 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. Applications: Markov Chains; Population Growth; Graph Theory; Error-Correcting Codes. Vignette: Robotics. Chapter Review. 4. EIGENVALUES AND EIGENVECTORS. Introduction: A Dynamical System on Graphs. Eigenvalues and Eigenvectors. Determinants. Exploration: Geometric Applications of Determinants. Eigenvalues and Eigenvectors of n x n Matrices. Similarity and Diagonalization. Iterative Methods for Computing Eigenvalues. Applications: Markov Chains; Population Growth; the Perron-Frobenius Theorem; Linear Recurrence Relations; Systems of Linear Differential Equations. Vignette: Ranking Sports Teams and Searching the Internet. Chapter Review. 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 Forms. Chapter Review. 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. Chapter Review. 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. Chapter Review. Appendix A: Mathematical Notation and Methods of Proof. Appendix B: Mathematical Induction. Appendix C: Complex Numbers. Appendix D: Polynomials.