Linear Algebra A Modern Introduction

In this innovative new Linear Algebra text, award-winning educator David Poole covers vectors and vector geometry first to enable students to visualize the mathematics while they are doing matrix operations. Rather than merely doing the calculations with no understanding of the mathematics, students will be able to visualize and understand the meaning of the calculations. By seeing the mathematics and understanding the underlying geometry, students will develop mathematical maturity and learn to think abstractly.

Preface

xi

To the Instructor

xix

To the Student

xxv

Vectors

1

(56)

Introduction: The Racetrack Game

1

(2)

The Geometry and Algebra of Vectors

3

(13)

Length and Angle: The Dot Product

16

(16)

Exploration: Vectors and Geometry

30

(2)

Lines and Planes

32

(16)

Exploration: The Cross Product

46

(2)

Code Vectors and Modular Arithmetic

48

(9)

Systems of Linear Equations

57

(74)

Introduction: Triviality

57

(1)

Introduction to Systems of Linear Equations

58

(9)

Exploration: Lies My Computer Told Me

65

(2)

Direct Methods for Solving Linear Systems

67

(22)

Exploration: Partial Pivoting

85

(1)

Exploration: Counting Operations---An Introduction to the Analysis of Algorithms

86

(3)

Spanning Sets and Linear Independence

89

(12)

Applications

101

(19)

Iterative Methods for Solving Linear Systems

120

(11)

Matrices

131

(116)

Introduction: Matrices in Action

131

(2)

Matrix Operations

133

(17)

Matrix Algebra

150

(10)

The Inverse of a Matrix

160

(19)

Subspaces, Basis, Dimension, and Rank

179

(22)

Introduction to Linear Transformations

201

(16)

Applications

217

(30)

Exploration: The LU Factorization

241

(6)

Eigenvalues and Eigenvectors

247

(103)

Introduction: A Dynamical System on Graphs

247

(2)

Introduction to Eigenvalues and Eigenvectors

249

(7)

Determinants

256

(30)

Exploration: Geometric Applications of Determinants

280

(6)

Eigenvalues and Eigenvectors of nXn Matrices

286

(10)

Similarity and Diagonalization

296

(11)

Iterative Methods for Computing Eigenvalues

307

(11)

Applications and the Perron-Frobenius Theorem

318

(32)

Orthogonality

350

(75)

Introduction: Shadows on a Wall

350

(3)

Orthogonality in Rn

353

(11)

Orthogonal Complements and Orthogonal Projections

364

(11)

The Gram-Schmidt Process and the QR Factorization

375

(14)

Exploration: The Modified QR Factorization

384

(3)

Exploration: Approximating Eigenvalues with the QR Algorithm

387

(2)

Orthogonal Diagonalization of Symmetric Matrices

389

(9)

Applications

398

(27)

Vector Spaces

425

(112)

Introduction: Magic Squares

425

(2)

Vector Spaces and Subspaces

427

(16)

Linear Independence, Basis, and Dimension

443

(16)

Change of Basis

459

(11)

Linear Transformations

470

(10)

The Kernel and Range of a Linear Transformation

480

(16)

The Matrix of a Linear Transformation

496

(24)

Exploration: Tilings, Lattices, and the Crystallographic Restriction

517

(3)

Applications

520

(17)

Distance and Approximation

537

(98)

Introduction: Taxicab Geometry

537

(2)

Inner Product Spaces

539

(18)

Exploration: Geometric Inequalities and Optimization Problems

552

(5)

Norms and Distance Functions

557

(18)

Least Squares Approximation

575

(23)

The Singular Value Decomposition

598

(22)

Applications

620

(15)

Appendix A Mathematical Notation and Methods of Proof

635

(10)

Appendix B Mathematical Induction

645

(8)

Appendix C Complex Numbers

653

(12)

Appendix D Polynomials

665

(12)

Appendix E Technology Bytes

677

(46)

Answers to Selected Odd-Numbered Exercises

723

(33)

Index

756

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Linear Algebra A Modern Introduction

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

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5% off 1 book, 7% off 2 books, 10% off 3+ books

Linear Algebra A Modern Introduction

Summary

Table of Contents

Supplemental Materials

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