Linear Algebra A Modern Introduction

by Poole, David

• ISBN13: 9780538735452
• ISBN10: 0538735457
• Edition: 3rd
• Format: Hardcover
• Copyright: 2010-05-25
• Publisher: Brooks Cole

Summary

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.

Table of Contents

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

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

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.

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

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

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

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

Mathematical Notation and Methods of Proof

Mathematical Induction

Complex Numbers

Polynomials

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