Rent 9780321287137 | Linear Algebra and Its Applications with CD-ROM, Update

Summary Table of Contents

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students'understanding of these concepts is vital to their master... MORE

Preface

A Note to Students

Linear Equations in Linear Algebra

(104)

... MORE

Introductory Example: Linear Models in Economics and Engineering

(104)

Systems of Linear Equations

(12)

Row Reduction and Echelon Forms

(14)

Vector Equations

(12)

The Matrix Equation Ax = b

(10)

Solution Sets of Linear Systems

(7)

Applications of Linear Systems

(8)

Linear Independence

(8)

Introduction to Linear Transformations

(9)

The Matrix of a Linear Transformation

(10)

Linear Models in Business, Science, and Engineering

(10)

Supplementary Exercises

102

(3)

Matrix Algebra

105

(80)

Introductory Example: Computer Models in Aircraft Design

105

(80)

Matrix Operations

107

(11)

The Inverse of a Matrix

118

(10)

Characterizations of Invertible Matrices

128

(6)

Partitioned Matrices

134

(8)

Matrix Factorizations

142

(10)

The Leontief Input-Output Model

152

(6)

Applications to Computer Graphics

158

(9)

Subspaces of Rn

167

(9)

Dimension and Rank

176

(7)

Supplementary Exercises

183

(2)

Determinants

185

(30)

Introductory Example: Determinants in Analytic Geometry

185

(30)

Introduction to Determinants

186

(6)

Properties of Determinants

192

(9)

Cramer's Rule, Volume, and Linear Transformations

201

(10)

Supplementary Exercises

211

(4)

Vector Spaces

215

(86)

Introductory Example: Space Flight and Control Systems

215

(86)

Vector Spaces and Subspaces

216

(10)

Null Spaces, Column Spaces, and Linear Transformations

226

(11)

Linearly Independent Sets: Bases

237

(9)

Coordinate Systems

246

(10)

The Dimension of a Vector Space

256

(6)

Rank

262

(9)

Change of Basis

271

(6)

Applications to Difference Equations

277

(11)

Applications to Markov Chains

288

(10)

Supplementary Exercises

298

(3)

Eigenvalues and Eigenvectors

301

(72)

Introductory Example: Dynamical Systems and Spotted Owls

301

(72)

Eigenvectors and Eigenvalues

302

(8)

The Characteristic Equation

310

(9)

Diagonalization

319

(8)

Eigenvectors and Linear Transformations

327

(8)

Complex Eigenvalues

335

(7)

Discrete Dynamical Systems

342

(11)

Applications to Differential Equations

353

(10)

Iterative Estimates for Eigenvalues

363

(7)

Supplementary Exercises

370

(3)

Orthogonality and Least Squares

373

(74)

Introductory Example: Readjusting the North American Datum

373

(74)

Inner Product, Length, and Orthogonality

375

(9)

Orthogonal Sets

384

(10)

Orthogonal Projections

394

(8)

The Gram--Schmidt Process

402

(7)

Least-Squares Problems

409

(10)

Applications to Linear Models

419

(8)

Inner Product Spaces

427

(9)

Applications of Inner Product Spaces

436

(8)

Supplementary Exercises

444

(3)

Symmetric Matrices and Quadratic Forms

447

Introductory Example: Multichannel Image Processing

447

Diagonalization of Symmetric Matrices

449

(6)

Quadratic Forms

455

(8)

Constrained Optimization

463

(8)

The Singular Value Decomposition

471

(11)

Applications to Image Processing and Statistics

482

(9)

Supplementary Exercises

491

The Geometry of Vector Spaces

Introductory Example: The Platonic Solids

Affine Combinations

Affine Independence

Convex Combinations

Hyperplanes

Polytopes

Curves and Surfaces

Optimization

Introductory Example: The Berlin Airlift

Matrix Games

Linear Programming---Geometric Method

Linear Programming---Simplex Method

Duality

Appendixes

A Uniqueness of the Reduced Echelon Form

(2)

B Complex Numbers

(6)

Glossary

(10)

Answers to Odd-Numbered Exercises

Index

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