These notes provide an integrated approach to linear algebra and ordinary differential equations based on computers --- in this case the software package MATLAB 1. We believe that computers can improve the conceptual understanding of mathematics --- not just enable the completion of complicated calculations. We use computers in two ways: in linear algebra computers reduce the drudgery of calculations and enable students to focus on concepts and methods, while in differential equations computers display phase portraits graphically and enable students to focus on the qualitative information embodied in solutions rather than just on developing formulas for solutions.
We develop methods for solving both systems of linear equations and systems of (constant coefficient) linear ordinary differential equations. It is generally accepted that linear algebra methods aid in finding closed form solutions to systems of linear differential equations. The fact that the graphical solution of systems of differential equations can motivate concepts (both geometric and algebraic) in linear algebra is less often discussed. These notes begin by solving linear systems of equations (through standard Gaussian elimination theory) and discussing elementary matrix theory. We then introduce simple differential equations --- both single equations and planar systems --- to motivate the notions of eigenvectors and eigenvalues. In subsequent chapters linear algebra and ODE theory are often mixed.
Regarding differential equations, our purpose is to introduce at the sophomore -- junior level ideas from dynamical systems theory. We focus on phase portraits (and time series) rather than on techniques for finding closed form solutions. We assume that now and in the future practicing scientists and mathematicians will use ODE solving computer programs more frequently than they will use techniques of integration. For this reason we have focused on the information that is embedded in the computer graphical approach. We discuss both typical phase portraits (Morse-Smale systems) and typical one parameter bifurcations (both local and global). Our goal is to provide the mathematical background that is needed when interpreting the results of computer simulation.
The integration of computers: Our approach assumes that students have an easier time learning with computers if the computer segments are fully integrated with the course material. So we have interleaved the instructions on how to use MATLAB with the examples and theory in the text. With ease of use in mind, we have also provided a number of preloaded matrices and differential equations with the notes. Any equation label in this text that is followed by an asterisk can be loaded into MATLAB just by typing the formula number. For the successful use of this text, it is important that students have access to computers with MATLAB and the computer files associated with these notes.
John Polking has developed an excellent graphical user interface for solving planar systems of autonomous differential equations called pplane5. We use pplane5 (and a companion code dfield5) instead of using the MATLAB native commands for solving ODEs. In these notes we also provide an introduction to pplane5 and the other associated software routines.
For the most part we treat the computer as a black box. We have not attempted to explain how the computer, or more precisely MATLAB, performs computations. Linear algebra structures are developed (typically) with proofs, while differential equations theorems are presented (typically) without proof and are instead motivated by computer experimentation.
There are two types of exercises included with most sections --- those that should be completed using pencil and paper (called Hand Exercises) and those that should be completed with the assistance of computers (called Computer Exercises).
Ways to use the text: We envision this course as a one-year sequence replacing the standard one semester linear algebra and ODE courses. There is a natural one semester Linear Systems course that can be taught using the material in this book. In this course students will learn both the basics of linear algebra and the basics of linear systems of differential equations. This one semester course covers the material in the first eight chapters. The Linear Systems course stresses eigenvalues and a baby Jordan normal form theory for matrices and culminates in a classification of phase portraits for planar constant coefficient linear systems of differential equations. Time permitting additional linear algebra topics from Chapters 9 and 10 may be included. Such material includes changes of coordinates for linear mappings, and orthogonality including Gram-Schmidt orthonormalization and least squares fitting of data.
We believe that by being exposed to ODE theory a student taking just the first semester of this sequence will gain a better appreciation of linear algebra than will a student who takes a standard one semester introduction to linear algebra. However, a more traditional Linear Algebra course can be taught by omitting Chapter 7 and de-emphasizing some of the material in Chapter 6. Then there will be time in a one semester course to cover a selection of the linear algebra topics mentioned at the end of the previous paragraph.
Chapters ??--?? We consider the first two chapters to be introductory material and we attempt to cover this material as quickly as we can. Chapter ?? introduces MATLAB along with elementary remarks on vectors and matrices. In our course we ask the students to read the material in Chapter ?? and to use the computer instructions in that chapter as an entry into MATLAB. In class we cover only the material on dot product. Chapter ?? explains how to solve systems of linear equations and is required for a first course on linear algebra. The proof of the uniqueness of reduced echelon form matrices is not very illuminating for students and can be omitted in classroom discussion. Sections whose material we feel can be omitted are noted by asterisks in the Table of Contents and Section ?? is the first example of such a section.
In Chapter ?? we introduce matrix multiplication as a notation that simplifies the presentation of systems of linear equations. We then show how matrix multiplication leads to linear mappings and how linearity leads to the principle of superposition. Multiplication of matrices is introduced as composition of linear mappings, which makes transparent the observation that multiplication of matrices is associative. The chapter ends with a discussion of inverse matrices and the role that inverses play in solving systems of linear equations. The determinant of a matrix is introduced and its role in determining matrix inverses is emphasized.
Chapter ?? This chapter provides a nonstandard introduction to differential equations. We begin by emphasizing that solutions to differential equations are functions (or pairs of functions for planar systems). We explain in detail the two ways that we may graph solutions to differential equations (time series and phase space) and how to go back and forth between these two graphical representations. The use of the computer is mandatory in this chapter. Chapter ?? dwells on the qualitative theory of solutions to autonomous ordinary differential equations. In one dimension we discuss the importance of knowing equilibria and their stability so that we can understand the fate of all solutions. In two dimensions we emphasize constant coefficient linear systems and the existence (numerical) of invariant directions (eigendirections). In this way we motivate the introduction of eigenvalues and eigenvectors, which are discussed in detail for matrices. Once we know how to compute eigenvalues and eigendirections, we then show how this information coupled with superposition leads to closed form solution to initial value problems, at least when the eigenvalues are real and distinct.
We are not trying to give a thorough grounding in techniques for solving differential equations in Chapter ??; rather we are trying to give an introduction to the ways that modern computer programs will represent graphically solutions to differential equations. We have included, however, a section on separation of variables for those who wish to introduce techniques for finding closed form solutions to single differential equations at this time. Our preference is to omit this section in the Linear Systems course as well as to omit the applications in Section ?? of the linear growth model in one dimension to interest rates and population dynamics.
Chapter ?? In this chapter we introduce vector space theory: vector spaces, subspaces, spanning sets, linear independence, bases, dimensions and the other basic notions in linear algebra. Since solutions to differential equations naturally reside in function spaces, we are able to illustrate that vector spaces other than arise naturally. We have found that, depending on time, the proof of the main theorem, which appears in Section ??, may be omitted in a first course. The material in these chapters is mandatory in any first course on linear algebra.
Chapters ?? and ?? At this juncture the text divides into two tracks: one concerned with the qualitative theory of solutions to linear and nonlinear planar systems of differential equations and one mainly concerned with the development of higher dimensional linear algebra. We begin with a description of the differential equations chapters.
Chapter ?? describes closed form solutions to planar systems of constant coefficient linear differential equations in three different ways: a direct method based on eigenvalues and eigenvectors, a method based on matrix exponentials, and a related method based on similarity of matrices. Each of these methods has its virtues and vices. Note that the Jordan normal form theorem for matrices is proved when discussing how to solve linear planar systems using similarity of matrices.
The qualitative description of phase portraits (saddles, sinks, sources, stability, centers, etc.) for planar linear systems is presented in Chapter ??. This description depends crucially on the linear algebra of matrices developed in Chapters ?? and ??.
Chapters ??, ??, ??, and ?? Chapter ?? discusses determinants, characteristic polynomials, and eigenvalues for matrices. Chapter ?? presents more advanced material on linear mappings including row rank equals column rank and the matrix representation of mappings in different coordinate systems. The material in Sections ?? and ?? could be presented directly after Chapter ??, while the material in Section ?? explains the geometric meaning of similarity.
Orthogonal bases and orthogonal matrices, least squares and Gram-Schmidt orthonormalization, and symmetric matrices are presented in Chapter ??. This material is very important, but is not required later in the text, and may be omitted.
The Jordan normal form theorem for matrices is presented in Chapter ??. Diagonalization of matrices with distinct real and complex eigenvalues is presented in the first two sections. The appendices, including the proof of the complete Jordan normal form theorem, are included for completeness and should be omitted in classroom presentations.
The Classroom Use of Computers At the University of Houston we use a classroom with an IBM compatible PC and an overhead display. Lectures are presented three hours a week using a combination of blackboard and computer display. We find it inadvisable to use the computer for more than five minutes at a time; we tend to go back and forth between standard lecture style and computer presentations. (The preloaded matrices and differential equations are important to the smooth use of the computer in class.)
We ask students to enroll in a one hour computer lab where they can practice using the material in the text on a computer, do their homework and additional projects, and ask questions of TA’s. Our computer lab happens to have 15 power macs. In addition, we ensure that MATLAB and the laode files are available on student use computers around the campus (which is not always easy). The laode files are on the enclosed CDROM; they may also be downloaded by using a web browser or by anonymous ftp.
Acknowledgements This course was first taught on a pilot basis during the 1995--96 academic year at the University of Houston. We thank the Mathematics Department and the College of Natural Sciences and Mathematics of the University of Houston for providing the resources needed to bring a course such as this to fruition. We gratefully acknowledge John Polking’s help in adapting his software for our use and for allowing us access to his code so that we could write companion software for use in linear algebra.
We thank Denny Brown for his advice and his careful readings of the many drafts of this manuscript. We thank Gerhard Dangelmayr, Michael Field, Michael Friedberg, Steven Fuchs, Kimber Gross, Barbara Keyfitz, Charles Peters and David Wagner for their advice on the presentation of the material. We also thank Elizabeth Golubitsky, who has written the companion Solutions Manual, for her help in keeping the material accessible and in a proper order. Finally, we thank the students who stayed with this course on an experimental basis and by doing so helped to shape its form.
Houston and Bayreuth Martin Golubitsky
May, 1998 Michael
Dellnitz