8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

1/17

second edition

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

2/17

This page intentionally left blank

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

3/17

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

4/17

LINEAR ALGEBRA

A Geometric Approach

second edition

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

5/17

This page intentionally left blank

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

6/17

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

7/17

LINEAR ALGEBRA

A Geometric Approach

second edition

Theodore Shifrin Malcolm R. Adams

University of Georgia

W. H. Freeman and Company NewYork

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

8/17

Publisher: Ruth Baruth Senior Acquisitions Editor: Terri Ward Executive Marketing Manager: Jennifer Somerville

Associate Editor: Katrina Wilhelm Editorial Assistant: Lauren Kimmich Photo Editor: Bianca Moscatelli Cover

and Text Designer: Blake Logan Project Editors: Leigh Renhard and Techsetters, Inc. Illustrations: Techsetters,

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

9/17

Inc. Senior Illustration Coordinator: Bill Page Production Manager: Ellen Cash Composition: Techsetters, Inc.

Printing and Binding: RR Donnelley

Library of Congress Control Number: 2010921838

ISBN-13: 978-1-4292-1521-3 ISBN-10: 1-4292-1521-6 2011, 2002 by W. H. Freeman and Company All rights reserved

Printed in the United States of AmericaFirst printing

W. H. Freeman and Company 41 Madison Avenue New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England

www.whfreeman.com

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

10/17

CONTENTS

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

11/17

Preface vii Foreword to the Instructor xiii Foreword to the Student xvii

Chapter 1 Vectors and Matrices 1

1. Vectors 1 2. Dot Product 18 3. Hyperplanes in Rn 28 4. Systems of Linear Equations and Gaussian

Elimination 36 5. The Theory of Linear Systems 53 6. Some Applications 64

Chapter 2 Matrix Algebra 81

1. Matrix Operations 81 2. Linear Transformations: An Introduction 91 3. Inverse Matrices 102 4.

Elementary Matrices: Rows Get Equal Time 110 5. The Transpose 119

Chapter 3 Vector Spaces 127

1. Subspaces of Rn 127 2. The Four Fundamental Subspaces 136 3. Linear Independence and Basis

143 4. Dimension and Its Consequences 157 5. A Graphic Example 170 6. Abstract Vector Spaces

176

v

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

12/17

vi

Contents

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

13/17

Chapter 4 Projections and Linear Transformations 191

1. Inconsistent Systems and Projection 191 2. Orthogonal Bases 200 3. The Matrix of a Linear

Transformation and the

Change-of-Basis Formula 208 4. Linear Transformations on Abstract Vector Spaces 224

Chapter 5 Determinants 239

1. Properties of Determinants 239 2. Cofactors and Cramers Rule 245 3. Signed Area in R2 and

Signed Volume in R3 255

Chapter 6 Eigenvalues and Eigenvectors 261

1. The Characteristic Polynomial 261 2. Diagonalizability 270 3. Applications 277 4. The Spectral

Theorem 286

Chapter 7 Further Topics 299

1. Complex Eigenvalues and Jordan Canonical Form 299 2. Computer Graphics and Geometry 314 3.

Matrix Exponentials and Differential Equations 331

For Further Reading 349 Answers to Selected Exercises 351 List of Blue Boxes 367 Index 369

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

14/17

PREFACE

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

15/17

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

16/17

8/10/2019 Linear Algebra .ageometric Approach W.H.freeman

17/17

We emphasize concepts and understanding why, doing proofs in the text and asking the student to do plenty in the

exercises. To help the student adjust to a higher level of mathematical rigor, throughout the early portion of the text

we provide blue boxes discussing matters of logic and proof technique or advice on formulating problem-solving

strategies. A complete list of the blue boxes is included at the end of the book for the instructors and the students

reference.

We use rotations, reflections, and projections in R2 as a first brush with the notion of a linear transformation whenwe introduce matrix multiplication we then treat linear transformations generally in concert with the discussion of

projections. Thus, we motivate the change-of-basis formula by starting with a coordinate system in which a

geometrically defined linear transformation is clearly understood and asking for its standard matrix.

We emphasize orthogonal complements and their role in finding a homogeneous system of linear equations that

defines a given subspace of Rn.

In the last chapter we include topics for the advanced student, such as Jordan canonical form, a classification of the

motions of R2 and R3, and a discussion of how Mathematica draws two-dimensional images of three-dimensional

shapes.

The historical notes at the end of each chapter, prepared with the generous assistance of Paul Lorczak for the first

edition, have been left as is. We hope that they give readers an idea how the subject developed and who the keyplayers were.

A few words on miscellaneous symbols that appear in the text: We have marked with an asterisk ( ) the problems

for which there are answers or solutions at the back of the text. As a guide for the new teacher, we have also marked

with a sharp () those theoretical exercises that are important and to which reference is made later. We indicate the

end of a proof by the symbol .

Significant Changes in the Second Edition

We have added some examples (particularly of proof reasoning) to Chapter 1 and streamlined the discussion in

Sections 4 and 5. In particular, we have included a fairly simple proof that the rank of a matrix is well defined and have

outlined in an exercise how this simple proof can be extended to show that reduced echelon form is unique. We have

also introduced the Leslie matrix and an application to population dynamics in Section 6.

We have reorganized Chapter 2, adding two new sections: one on linear transfor- mations and one on elementary

matrices. This makes our introduction of linear transformations more detailed and more accessible than in the first

edition, paving the way for continued exploration in Chapter 4.

We have combined the sections on linear independence and basis and noticeably streamlined the treatment of the

four fundamental subspaces throughout Chapter 3. In particular, we now obtain all the orthogonality relations among

these four subspaces in Section 2.

We have altered Section 1 of Chapter 4 somewhat and have completely reorga- nized the treatment of the

change-of-basis theorem. Now we treat first linear maps T : Rn Rn in Section 3, and we delay to Section 4 the

general case and linear maps on abstract vector spaces.

We have completely reorganized Chapter 5, moving the geometric interpretation of the determinant from Section 1 to

Section 3. Until the end of Section 1, we have tied the computation of determinants to row operations only, proving at

the end that this implies multilinearity.