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8/10/2019 Linear Algebra .ageometric Approach W.H.freeman
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second edition
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LINEAR ALGEBRA
A Geometric Approach
second edition
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LINEAR ALGEBRA
A Geometric Approach
second edition
Theodore Shifrin Malcolm R. Adams
University of Georgia
W. H. Freeman and Company NewYork
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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,
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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
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CONTENTS
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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
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vi
Contents
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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
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PREFACE
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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.