Linear Algebra and Differential Equations: Pearson New International Edition

481 Pages·29.865 MB·English

by  Peterson, Gary L

Pearson New International Edition
Linear Algebra & Differential Equations

Gary L. Peterson James S. Sochacki
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ISBN 10: 1-292-04273-7
ISBN 13: 978-1-292-04273-2
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ISBN 10: 1-292-04273-7
ISBN 10: 1-269-37450-8
ISBN 13: 978-1-292-04273-2
ISBN 13: 978-1-269-37450-7
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Preface
The idea of an integrated linear algebra and differential equations course is an attractive
one to many mathematics faculty. One significant reason for this is that, while linear
algebra has widespread application throughout mathematics, the traditional one-term
linear algebra course leaves little time for these applications-especially those deal
ing with vector spaces and eigenvalues and eigenvectors. Meanwhile, linear algebra
concepts such as bases of vector spaces and eigenvalues and eigenvectors arise in the
traditional introductory ordinary differential equations course in the study of linear dif
ferential equations and systems of linear differential equations. We began offering such
an integrated course at James Madison University several years ago. Our experiences in
teaching this course along with our unhappiness with existing texts led us to write this
book.
As we tried various existing texts in our course, we realized that a better organization
was in order. The most typical approach has been to start with the beginning material of
an introductory ordinary differential equations text, thcn cover the standard material of a
linear algebra course, and finally return to the study of differential equations. Because of
the disjointed nature and inefficiency of such an arrangement, we started to experiment
with different arrangements as we taught out of these texts that evolved to the one
reflected by our table of contents. Preliminary drafts of our book with this arrangement
have been successfully class tested at James Madison University for several years.
TEXT ORGANIZATION
To obtain a better integrated treatment of linear algebra and differential equations, our
arrangement begins with two chapters on linear algebra, Chapter 1 on matrices and
determinants and Chapter 2 on vector spaces. We then turn our attention to differential
equations in the next two chapters. Chapter 3 is our introductory chapter on differential
equations with a primary focus on first order equations. Students have encountered a
good bit of this material already in their calculus courses. Chapter 4 then links together
iii
iv Preface
the subject areas of linear algebra and differential equations with the study of linear
differential equations. In Chapter 5 we return to linear algebra with the study of linear
transformations and eigenvalues and eigenvectors. Chapter 5 then naturally dovetails
with Chapter 6 on systems of differential equations. We conclude with Chapter 7 on
Laplace transforms, which includes systems of differential equations, Chapter 8 on power
series solutions to ditlerential equations, and Chapter 9 on inner product spaces. The
following diagram illustrates the primary dependencies of the chapters of this book.
Chapter 1
Matrices and Determinants
Chapter 3
Chapter 2 First Order Ordinary
Vector Spaces Differential Equations
Chapter 5
Chapter 4
Linear Transformations
Linear Differential
and Eigenvalues
and Eigenvectors
Chapter 6
Chapter 7
Systems of Differential
The Laplace Transform
Equations
Chapter 9 Chapter 8
Inn~r Product Spaces Power Series
Solutions
PRESENTATION
In writing this text, we have striven to write a book that gives the student a solid foundation
in both linear algebra and ordinary differential equations. We have kept the writing style
to the point and student friendly while adhering to conventional mathematical style.
Theoretic developments are as complete as possible. Almost all proofs of theorems have
been included with a few exceptions, such as the proofs of the existence and uniqueness of
solutions to higher order and systems oflinear differential equation initial value problems
in Chapters 4 and 6 and the existence of Jordan canonical form in Chapter 5. In a number
of instances we have strategically placed those proofs that, in our experience, many
instructors are forced to omit so that they can be skipped with minimum disruption to the
Preface v
flow of the presentation. We have taken care to include an ample number of examples,
exercises, and appl ications in this text. For motivational purposes, we begin each chapter
with an introduction relating its contents to either knowledge already possessed by the
student or an application.
COURSE ARRANGEMENTS
While our motivation has been to write a text that would be suitable for a one-term
course, this book certainly contains more than enough material for a two-term sequence
of courses on linear algebra and differential equations consisting of either two integrated
courses or a linear algebra course followed by a differential equations course. For those
teaching a one-term course, we describe our integrated linear algebra and differential
equations course at James Madison University. This course is a four-credit one-semester
course with approximately 56 class meetings and is centered around Chapters 1-6.
Most students taking this course either have completed the calculus sequence or are
concurrently in the final semester of calculus. We suggest instructors of our course
follow the following approximate 48-day schedule:
Chapter 1 (10 days): 2 days each on Sections 1.1-1.3, 1 day on Section 1.4,
3 days on Sections 1.5 and 1.6
Chapter 2 (9 days): 2 days each on Sections 2.1-2.4 and I day on Section 2.5
Chapter 3 (7 days): 4 days on Sections 3.1-3.4, 2 days on Section 3.6, and I day
on Section 3.7
Chapter 4 (9 days): 2 days on each of Sections 4.1-4.3, 1 day on Section 4.4, and
2 days on Section 4.5
Chapter 5 (7 days): 2 days on Sections 5.1 and 5.2,2 days each on Sections 5.3
and 5.4, and I day on Section 5.5
Chapter 6 (6 days): 1 day on Section 6.1,2 days on Section 6.2,1 day on
Sections 6.3 and 6.4, and 2 days on Sections 6.5 and 6.6.
After adding time for testing, this schedule leaves an ample number of days to cover
selected material from Chapters 7-9 or to devote more time to material in Chapters 1-
6. Those interested in a two-term sequence of integrated linear algebra and differential
equations courses should have little difficulty constructing a syllabus for such a sequence.
A sequence of courses consisting of a linear algebra course followed by a differential
equations course can be built around Chapters 1,2,5, and 9 for the linear algebra course
and Chapters 3, 4, 6, 7, and 8 for the differential equations course.
MATHEMATICAL SOFTWARE
This text is written so that it can be used both by those who wish to make use of
mathematical software packages for linear algebra and differential equations (such as
Maple or Mathematica) and by those who do not wish to do so. For those who wish
to do so, exercises involving the use of an appropriate software package are included
throughout the text. To facilitate the use of these packages, we have concluded that it
is best to include discussions with a particular package rather than trying to give vague
generic descriptions. We have chosen Maple as our primary accompanying software
vi Preface
package for this textbook. For those who wish to use Mathematica or MATLAB instead
of Maple, a supplementary Technology Resource Manual on the use of these packages
for this book is available from Addison-Wesley (ISBN: 0-201-75815-6). Those wishing
to usc other software packages should have little difficulty incorporating the use of these
other packages. In a few instances, instructions on how to use Maple for Maple users
appear within the text. But most of the time we simply indicate the appropriate Maple
commands in the exercises with the expectation that either the student using Maple will
consult Maple's hclp menu to see how to use the commands or the instructor will add
supplements on how to apply Maple's commands.
ACKNOWLEDGMENTS
Tt is our pleasure to thank the many people who have helped shape this book. We arc
grateful to the students and faculty at James Madison University who put up with the
preliminary drafts of this text and offered numerous improvements. We have received
many valuable suggestions from our reviewers, including:
Paul Eenigenburg, Western Michigan University
Gary Howell, Florida Institute of Technology
David Johnson, Lehigh University
Douglas Meade. University of South Carolina
Ronald Miech, University of California, Los Angeles
Michael Nasab, Long Beach City College
Clifford Queen, Lehigh University
Murray Schechter, Lehigh University
Barbara Shipman, University of Texas at Arlington
William Stout, Salve Regina University
Russell Thompson, Utah State University
James Turner, Calvin College (formerly at Purdue University)
Finally, we wish to thank the staff of and individuals associated with Addison-Wesley in
cluding sponsoring cditor Laurie Rosatone, project manager Jennifer Albanese, assistant
editors Ellen Keohane and Susan Laferriere, production supervisor Peggy McMahon,
copyediting and production coordinator Barbara Pendergast, tcchnical art supervisor Joc
Vetere, and mathematics accuracy checker Paul Lorczak whose efforts kept this project
on track and brought it to fruition. To all of you, we owe a debt of gratitude.
G. L. P.
J. S. S.
Harrisonburg, VA
Contents
1
Matrices and Determinants 1
1.1 Systems of Linear Equations 2
1.2 Matrices and Matrix Operations 17
1.3 Inverses of Matrices 28
1.4 Special Matrices and Additional Properties of Matrices 37
1.5 Determinants 43
1.6 Further Properties of Determinants 51
1.7 Proofs of Theorems on Determinants 58
2
Vector Spaces 65
2.1 Vector Spaces 66
2.2 Subspaces and Spanning Sets 74
2.3 Linear Independence and Bases 83
2.4 Dimension; Nullspaee, Row Space, and Column Space 95
2.5 VVronskians 106
vii
viii Contents
3
First Order Ordinary Differential Equations III
3.1 Introduction to Differential Equations 112
3.2 Separable Differential Equations 120
3.3 Exact Differential Equations 124
3.4 Linear Differential Equations 130
3.5 More Techniques for Solving First Order Differential Equations 136
3.6 Modeling with Differential Equations 144
3.7 Reduction of Order 153
3.8 The Theory of First Order Differential Equations 157
3.9 Numerical Solutions of Ordinary Differential Equations 168
4
Linear Differential Equations 179
4.1 The Theory of Higher Order Linear Differential Equations 179
4.2 Homogeneous Constant Coefficient Linear Differential Equations 189
4.3 The Method of Undetermined Coefficients 203
4.4 The Method of Variation of Parameters 211
4.5 Some Applications of Higher Order Differential Equations 217
5
Linear Transformations and Eigenvalues
and Eigenvectors 231
5.1 Linear Transformations 231
5.2 The Algebra of Linear Transformations; Differential Operators
and Differential Equations 245
5.3 Matrices for Linear Transformations 253
5.4 Eigenvalues and Eigenvectors of Matrices 269
5.5 Similar Matrices, Diagonalization, and Jordan Canonical Form 278
5.6 Eigenvectors and Eigenvalues of Linear Transformations 287
Contents ix
6
Systems of Differential Equations 293
6.1 The Theory of Systems of Linear Differential Equations 295
6.2 Homogeneous Systems with Constant Coefficients:
The Diagonalizable Case 302
6.3 Homogeneous Systems with Constant Coefficients:
The Nondiagonalizable Case 312
6.4 Nonhomogeneous Linear Systems 315
6.5 Converting Differential Equations to First Order Systems 319
6.6 Applications Involving Systems of Linear Differential Equations 322
6.7 2 x 2 Systems of Nonlinear Differential Equations 334
7
The Laplace Transform 345
7.1 Definition and Properties of the Laplace Transform 345
7.2 Solving Constant Coefflcient Linear Initial Value Problems
with Laplace Transforms 352
7.3 Step Functions, Impulse Functions, and the Delta Function 356
7.4 Convolution Integrals 366
7.5 Systems of Linear Differential Equations 370
8
Power Series Solutions to Linear Differential
Equations 375
8.1 Introduction to Power Series Solutions 376
8.2 Series Solutions for Second Order Linear DifIerential Equations 384
8.3 Euler Type Equations 393
8.4 Series Solutions Near a Regular Singular Point 397