INSTRUCTOR’S
MATLAB® MANUAL
JEREMY R. CASE
Taylor University
JANE DAY
San Jose State University
LINEAR ALGEBRA
AND ITS APPLICATIONS
FOURTH EDITION
David C. Lay
University of Maryland
The author and publisher of this book have used their best efforts in preparing this book. These efforts include
the development, research, and testing of the theories and programs to determine their effectiveness. The author
and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the
documentation contained in this book. The author and publisher shall not be liable in any event for incidental or
consequential damages in connection with, or arising out of, the furnishing, performance, or use of these
programs.
iii
Contents
1 Getting Started …………………………………………… 1
Prepare …………………………………………………………………………………1
Obtain an Educational License ………………………………………………..1
Prepare Student Computer Lab Instructions ……………………………..2
2 Planning the Course …………………………………… 2
Allow Time for Planning and Adjusting Plans …………………………..2
Consider Purposes for Computer Assignments ………………………….3
Decide Emphasis on Computer Work ……………………………………….4
Design Computer Assignments ……………………………………………….4
3 Using Software for Demonstrations …………….. 8
4 Downloading M-Files from the Web ……………. 9
5 Computer Projects ……………………………………. 10
General Information ……………………………………………………………..10
Partners ………………………………………………………………………………10
Notes about the Individual Projects ………………………………………..10
6 Overview of the Case Studies/Applications … 14
Case Studies ………………………………………………………………………..14
Application Projects ……………………………………………………………..14
7 References ………………………………………………… 16
Sample Computer Projects
iv
Forward
This manual is for any instructor who is using MATLAB and Linear Algebra and Its Applications together for the
first time. It will greatly simplify your task of combining MATLAB with the text, because it is written by a colleague
who has already tried out the materials with considerable success. This manual carefully describes everything you
need to know about planning and conducting the course.
I am pleased with the work of the author, Professor Jeremy Case, who revised this MATLAB manual. He
substantially reorganized and edited the main part, adding his own perspective based on his use of MATLAB and the
Introduction
It is a privilege to be a part of this supplemental MATLAB manual to accompany the fourth edition of Linear Algebra
and Its Applications by David C. Lay. The following materials and projects are essentially the work of Jane M. Day
of San Jose State University. She was the primary author for the MATLAB manuals to accompany Lay’s Linear
Algebra prior to the third edition. I have made only minor editing to her work.
The purpose of this manual is to help you integrate computer exercises into your linear algebra course using
MATLAB. MATLAB and Linear Algebra work very well together, and your students have an excellent opportunity to
gain a better understanding of the material.
1. Getting Started
PREPARE
This manual assumes that you will be using MATLAB, but there are other excellent software packages or
calculators you could use. Maple and Mathematica are examples of other mathematical software packages
conducive to linear algebra. Calculators with matrix capabilities such as those made by Hewlett Packard and Texas
Instruments are other possibilities. Each of the technologies listed here has a manual to accompany Lay’s book and
is available from Addison-Wesley. The exercises in Lay’s text are written so that any appropriate software or
calculator may be used.
This manual assumes that in addition to MATLAB you will use Laydata4 Toolbox, which is a collection of M-files
that can be downloaded from MyMathLab or accessed through pearsonhighered.com/lay. These M-files will need
to be made accessible to your students. It is also a good idea to use the Study Guide, a supplement to the text, to
accompany this manual. The text, the Study Guide, and Laydata4 Toolbox are highly coordinated to work together.
For the students, the Study Guide is the primary support for the use of technology during the semester. In addition to
the MATLAB boxes at the ends of many sections, the Appendix, “Getting Started with MATLAB,” provides the
OBTAIN AN EDUCATIONAL LICENSE
Usually the most cost effective way to provide MATLAB to students is for your school to buy an educational site
license. You can use any version of MATLAB for Lay’s [M] exercises and for most of the projects.
ORDER STUDENT MATLAB AND STUDY GUIDES EARLY
Ask your bookstore to stock Lay’s Study Guide with the textbook. Our institution has an educational site license for
MATLAB, but if yours does not, request that the Student Edition of MATLAB [10] be made available as well. The
latest edition of Student MATLAB is usually what they will get. Some students may have access to earlier editions,
and those will work fine. Student MATLAB costs about $100 and includes a good User‘s Guide. It is identical to
professional MATLAB except in a few ways that rarely affect students’ use.
INSTALL LAYDATA4 TOOLBOX BEFORE CLASSES START
2 Planning the Course
You should check the status of your software and M-files before each term begins. One year, our institution changed
operating systems between semesters. What had worked one month previously no longer worked, and I spent the
first week scrambling to correct it. It got the class off on the wrong foot, and it took longer for them to feel
comfortable with the technology. It taught me to never take for granted what will work as computers are upgraded.
OBTAIN DATA FOR CASE STUDIES AND APPLICATION PROJECTS
In addition to the hard copy projects at the end of this manual, there are Case Studies and Application Projects
available from the Web. The Case Studies expand topics introduced at the beginning of each chapter in Lay’s
textbook and use real-world data. The Application Projects either extend existing topics in the text or introduce new
PREPARE STUDENT COMPUTER LAB INSTRUCTIONS
On the first day of classes, students need information about how you plan to use MATLAB in the course and how
they can access the program and appropriate data. List the Study Guide as the “lab manual” for the course. With the
Study Guide in hand, students rarely will need more documentation for the course other than MATLAB’s help
command and your local computer procedures. You can prepare a sheet to hand out, or put the information on a web
page, or do both. Here are some facts that students may need:
• Location of campus computer lab facilities.
• Hours and days when the labs are available for student use.
2. Planning the Course
ALLOW TIME FOR PLANNING AND ADJUSTING PLANS
It would be very good to have some release time the first time you try using a significant number of computer
exercises. However, some institutions like mine cannot always provide such release time. Pressed for time, I found
the projects in this manual to be very helpful the first time I taught linear algebra.
Planning the Course 3
CONSIDER PURPOSES FOR COMPUTER ASSIGNMENTS
As you consider your students’ interests and begin to appreciate the potential of computer exercises, decide what
purposes are most appropriate for your class. Here are some possible ones:
2. To reinforce understanding of concepts and theory
4. To explore and conjecture
6. To learn something new
8. To practice routine calculations
10. To introduce MATLAB for later courses
The first four purposes listed are the most important to me, but all of these reasons have merit and are addressed in
various projects.
Applications provide motivation as to why one should learn the material. For many of my students, they seem more
likely to forget material that they do not find interesting or that they cannot conceive of applying it in the future.
You can expose your students to a variety of applications using the case studies in the book. Such examples are
natural topics for computer exercises because applications are more interesting when the data are not trivial.
The second goal is very important. People tend to misuse theory when they do not understand it, so you might stress
the mastery of the big ideas such as linear independence, span, basis, dimension, eigenvalues and orthogonality. I
I consider the third and fourth purposes part of the broader scheme of mathematics. I will then add some questions to
the projects asking students to extend the results. These are sometimes difficult to grade particularly if you have a
1. Professionally written matrix software gives good answers to most problems, but there is almost
always some error.
3. Some problems are inherently difficult to solve accurately even with the best algorithms. So users
should never ignore warnings from professional software.
For students who want to know more, there are several well-written contemporary introductions to numerical linear
algebra. George Forsythe’s paper [4] is a classic. It is remarkable how clearly he articulated the inevitable pitfalls of
floating point matrix computations so early.
4 Planning the Course
DECIDE EMPHASIS ON COMPUTER WORK
Computer projects are good vehicles for introducing simple applications, which are important for the majority of our
students. One important reason for having linear algebra students work with professional software like MATLAB is
that they need to know such software exists. In the workplace, they will be using professional software that will be
far faster and more accurate than code based on algorithms alone. At the same time, I feel it is important for them to
learn the basic algorithms of linear algebra because those enhance understanding of concepts.
How much you will emphasize computer assignments, applications, and theory depends on your personal situation.
Your teaching style, your course objectives, and the specific needs at your school are all factors that must be given
consideration. I suggest you look at book [2] which greatly helped me to become aware of issues I had not
previously considered and to navigate these issues for myself.
DESIGN COMPUTER ASSIGNMENTS
The first time you use computer assignments, you should probably proceed with some caution. Evaluate the
effectiveness and difficulty of each assignment before making another. Various details can require more attention
than you might expect, especially at the beginning of the semester. For instance, you may find that access to the
computer labs is inadequate, or a student who buys software has trouble installing it. Equipment has a way of
breaking down when you need it most. It would be wise to assume “If it can go wrong, it will” and then be
pleasantly surprised if things go smoothly. On the other hand, technology is so ubiquitous that I am finding fewer
and fewer problems each year. (It could also be that I know now by experience which IT person to contact.)
It is important to give an easy computer assignment early and collect it to get students started and to help you
evaluate how they react to computer use. I suggest you ask them to work through “Getting Started” during the first
few days. It will be good for them to know what topics are discussed there even if they don‘t understand all the
students select 2 or 3 more for extra credit, which seems a reasonable way to address the variety of student interests.
It is a good idea to discuss each project briefly before assigning it and again when handing the papers back to be
sure everyone got the point. A very real danger in computer projects is that students push the buttons and miss the
obvious points in understanding. The projects here are written like lab forms and from my experience are pretty easy
to grade. Student graders, if available, can be used to grade the projects to save you time.
Planning the Course 5
I encourage students to work in groups. I am almost convinced that students do better with technology when they
have to figure it out for themselves rather than when it is explained to them. Furthermore, there are the MATLAB
boxes and the MATLAB Appendix in the Study Guide for them to use as a resource. Since it is unlikely that
ANTICIPATE COMMON DIFFICULTIES
Most students today are very computer literate. However there are still a few techno-phobes in every class. Their
problems are primarily not with MATLAB, but rather with the interface between it and the outside environment,
such as saving, editing and printing files. These projects were designed to minimize these issues because these
CONSIDER CLASSROOM DEMONSTRATIONS
Most schools have permanent or portable demonstration units in the classroom consisting of a computer and a
multimedia projector. The portable ones have the inconvenience of setting them up and down before class, but they
do work. For linear algebra purposes, a color display is desirable because of the demonstrations related to graphics.
While it is very difficult at our institution to reserve a computer lab on a regular basis, I can use such a projection
unit with a computer and a camera on a calculator for classroom demonstrations. This has benefited my classes so
much that I resent being assigned a classroom with just a chalkboard for my other courses. If I understand the
educational research correctly, students learn much more by seeing the material in a variety of contexts. I sometimes
give students “control” of the computer to break up the routine in class. They often can calculate faster than me, and
the class occasionally comes up with new insights since the method of using MATLAB is slightly different.
6 Planning the Course
DECIDE HOW YOU WILL TEST STUDENTS
How students should be assessed on exams is always a critical issue for an instructor. One major consideration is
whether to allow MATLAB on exams. If MATLAB is unavailable, you will need to decide if graphing calculators
will be permitted. You obviously will have to consider the profiles of your students and the test conditions.
My approach is that students should primarily be tested on conceptual ideas rather than computations. There are
other settings, such as homework, where computational skills can be assessed. I allow TI-84 and TI-89 graphing
calculators on tests because the use of MATLAB for me is not practical. I still try to minimize the need for a
calculator unless I give an untimed test or I link the data directly to their calculators. (I do not allow other hand held
devices although connectivity to the “outside world” will increasingly become an interesting and challenging issue.)
As far as the type of questions to include on an exam, I like test questions that can be quickly done if one takes the
appropriate perspective and thinks about the issues involved. I also value questions that require the synthesis of a
1. The following matrices are row equivalent:
1232
1208
2456
A
−−
⎡
⎤
⎢
⎥
=− −
⎢
⎥
⎢
⎥
−−
⎣
⎦
,
120 8
001 2
000 0
R
−
⎡
⎤
⎢
⎥
=−
⎢
⎥
⎢
⎥
⎣
⎦
.
Write the general solution to A=x0. Write a particular solution to A=x0. Consider the matrix transformation
A
→xx; is it 1-1? onto? Explain answers. Find a basis for the column space of A and a basis for the null space of
3. A certain population of owls feeds almost exclusively on wood rats. Letting ()ok and ()rk denote the
number in each population in year k, a biologist estimates that (1).5().05()ok ok rk+= + and
(1) .9()50()rk ok rk+=− + . Write the matrix that describes the interaction of these two populations from year k to
year k+1.
Assume the pattern described will continue in the future. Don’t calculate, but instead answer in words:
4. Consider the vectors v1 = (1,1,1,1), v2 = (2,1,0,-3), and v3 = (-1,2,0,0).
(a) Which pairs of these vectors are orthogonal to each other? Show work.
5. Let A be the nn× matrix in which each entry is 1. Justify your answers to the following questions. For (a)-
(c), think, don’t calculate! A very little calculation will be needed for (d).
(a) There are two distinct eigenvalues of A. What are they?
(b) What is dim(Nul(A))?
(c) What is the characteristic polynomial of A?
(d) What is a basis for each eigenspace of A?
6. Suppose A is an invertible matrix, and x is an eigenvector of A.
(a) Which of the following matrices also have x as an eigenvector? Circle the ones that do:
.
7. Let A be an nn× matrix. In each part below, circle the one best possible expression to complete the
sentence truthfully. Only that one best choice will be counted correct:
(a) Suppose A has four distinct eigenvalues. Then A (will) (will not) (could but doesn’t have to) have
four independent eigenvectors.
BE CREATIVE
The very existence of powerful and accessible matrix utilities raises questions about what topics to emphasize, what
skills students need to learn, and what style of teaching is best. These issues are not easily resolved. Many have been
influenced by the constructivist theory that students learn best when making connections to what they already know
or what they want to know. As mentioned before, I have used certain projects for certain majors and had them
expand on those ideas by having them develop an education lesson plan or by writing a report after doing some
independent reading. Each year I try to use more group work in my courses so that students feel more involved in
the course. I encourage my students to study and to work on problems together, and I think working together lowers
the frustration level with regards to computers.
you are doing to allow them to appreciate your subject matter.
3. Using Software for Demonstrations
Most of the M-files in Laydata4 Toolbox simply contain data for exercises and projects, and this can be helpful for
demonstrations as discussed on page 5. In addition, there are a few files that are called the “special functions,”
which do particular kinds of calculations or graphing. They were written for various exercises and projects but they
can also be effective for occasional demonstrations. I encourage you to experiment with them early so you’ll know
about them when an occasion arises where they might be helpful. Here is a list of these special functions with a few
comments:
Special Function Description
replace, swap, scale Single row operations
gauss, bgauss Sweep out specified columns
The five functions below produce simple but effective graphics.
seesum, seeprod, seecom Visualize vector arithmetic
drawpoly Draw polygons
singvec Search visually for singular vectors
The function randomint allows you to specify size and rank, and is very useful for generating quick, clear
examples. The command randomint(5,4,2) will create a 54×matrix of rank 2.
randomint Create random integer matrices
The simple way to find out how to use any MATLAB function is with help. For example, at the MATLAB prompt,
type help replace or help seesum.
Although all these special functions are very nice, at some point emphasize that they were developed for educational
purposes and should not be used for professional applications. If such a need arises, they should use professionally
written software that employs the most sophisticated and efficient algorithms known. For example, in MATLAB one
4. Downloading M-files from the Web
To get Laydata4 Toolbox files, go to
www.pearsonhighered.com/lay
Follow the on-screen directions to obtain the version of Lay’s files you want.
If you have used a Toolbox from an earlier edition of the book, you will want to delete the folder. In earlier editions
of the text, the Toolbox was named Laydata and will likely be found in the main MATLAB folder.
Once you have the files, you will want to decompress them and make them accessible to the working path so that
MATLAB knows where to find them. To avoid having to type a possibly complicated path to the correct folder,
create an empty folder name laydata4 inside the main MATLAB folder. It is a good idea to use lowercase letters
since MATLAB is case sensitive. For example, on a PC the file should be created in c:\matlab\ (or whatever your
working MATLAB path is). With the folder already named, navigation is easier as you move through the directory
tree and decompress the downloaded files into the appropriate folder.
5. Computer Projects
GENERAL INFORMATION
The projects are intended to enrich and expand the material in Lay’s text. They are independent of each other. Each
one begins by stating a purpose, the prerequisite sections from text, and the MATLAB functions used. On the line
listing the MATLAB functions, the commands inherent to MATLAB are listed first and are followed by a
semicolon. The functions and data files from Laydata4 Toolbox follow the semicolon. You may copy and use the
PARTNERS
I encourage but do not require students to do their computer work with a partner. This helps the students work
through some of the computer issues together and cuts down on computer frustration. It also reduces my grading.
Most of my classes do well in pairing up, but some students need help finding a partner even after the first couple of
weeks.
NOTES ABOUT THE INDIVIDUAL PROJECTS
Here are a few comments about each project. The symbol R means the project is especially recommended because
of its value. I usually grade each project out of 10 points and am generally more lenient with my grading than on
other homework.
Getting Started With MATLAB. This is long and it is not necessary to do it all, but it can be helpful for
novices and for reference.
R Practice Row Reduction (5). This is easy and could be assigned soon after students have learned to do row
operations by hand. They will practice doing them with Lay’s functions replace, scale, and swap.
Computer Projects 11
Reduced Echelon Form and ref (5). This is easy and shows students how to row-reduce a matrix using
gauss and ref. They also see that roundoff error can cause ref to produce the wrong answer, by experimenting
with different values for the tolerance in ref. Although the command ref can be introduced as early as Section
1.2, the text emphasizes echelon form rather than reduced echelon form until Section 1.5. The Study Guide uses
gauss, swap, and scale for row operations until Section 4.3 (and 2.9).
1.9)
Population Migration (5). Students like this, especially the plotting. Before assigning this one and going
over the city-suburb example in the text, you might ask your students what will happen if this pattern of migration
persists. Will everyone move to the suburbs? Have them calculate k
x for some large values of k and report back at
2.1)
The Adjacency Matrix of a Graph (10). This is a more sophisticated look at matrix multiplication.
Students examine very carefully how an entry of 2
A
is calculated. They must also create a definition, which is a
new kind of exercise for most of them. Before assigning this project, I recommend you discuss graphs a little and
2.3)
12 Computer Projects
R Roundoff Error in Matrix Calculations (5). Students use backslash, ref and inv to solve 8 linear
systems three different ways. They see that the different algorithms give somewhat different answers in every case
and very different answers for the poorly conditioned systems. They see definitions of floating point notation,
residual vector, condition, and Hilbert matrix; learn to watch for warnings; and use norm. There are brief
discussions of condition number and the algorithms used in the three methods. The illustrated computational
realities are important since many students will do matrix calculations in scientific applications. (Section 2.3)
Partitioned Matrices (5). This is an important topic, as partitioned matrices are used frequently in
Schur Complement (5). This is a nice application of partitioning. Students learn three ways to calculate a
Schur complement, including using row operations. The extra credit question is challenging. (Section 2.4)
LU Factorization (5). This explains how the LU factorization algorithm in Section 2.5 differs from
MATLAB’s lu function, and provides practice using both. (Section 2.5)
Homogeneous Coordinates for Computer Graphics (5). This uses drawpoly. Homogeneous
coordinates for 2
and how they can be manipulated with 33×matrices are novel ideas to most students.
Computer science majors especially like this. (Section 2.7)
Subspaces (20). Span is a hard concept for many students, and this has proved to be the most challenging
project. You can pair up the students and assign each group a different pair of matrices, A and B. Each matrix A is
54×, B is 55×, and both have rank 4. There are 25 pairs of integer matrices in the file submats. The first 17
and then conclude that the two sets must be the same, but few students seem to think of doing that. Instead they
Computer Projects 13
explain that Col B is inside Col A and has the same dimension so by Theorem 15, a basis for Col B must span Col A.
This project could be assigned after Section 4.6 (or Section 2.9 if you cover that instead).
Markov Chains and Long-Range Predictions (5-10 points, depending on how much help is given ahead
of time). This is fun and good motivation for eigenvalues and eigenvectors. If you ask everyone to do this project,
you should give a brief introduction to Markov processes. (Section 4.9)
The extra credit question asks users to verify theoretically what they have seen experimentally. It is
challenging but many hints are given. The idea for this question is due to Andre Weideman and used with his
permission. Techniques from calculus can be used to show that the stage matrix will always have its dominant
eigenvalue real and positive and be diagonalizable, and then one can derive a formula for the critical value of t.
(Sections 5.5 and 5.6)
QR Factorization (5). This is not hard and should be of interest to many students, since QR factorizations
are widely used in practice. Students will learn how MATLAB’s qr function differs from the QR factorization
developed in the text and also will verify the connection beween QR factorization and the Gram Schmidt Process.
(Section 6.4)
R Least-Squares Solutions and Curve Fitting (5). This is easy and can be done early in Chapter 6 to
motivate the ideas. The text algorithm is used to calculate coefficients for the least squares line, quadratic and cubic
curves; norm is used to calculate least squares error, and plot is used to graph the data and the curves. An end
note describes how to use polyval and polyfit, which are MATLAB functions that can do most of the work
for you.
In question 2(b), most students guess 3
()vel and say the error is smaller for the cubic than for the line or
quadratic, and that is acceptable to some including me. However, the correct answer is that drag depends on 2
()vel
6. Overview of the Case Studies and Application Projects
The following case studies and application projects are available from the website which accompanies the text. An
icon in the text refers the reader to these resources. The case studies amplify the opening vignette of each chapter
and provide exercises based upon the topic mentioned in the vignette. The application projects highlight applications
of linear algebra and direct students through a sequence of exercises on that application. Many of these resources use
real world data. This data, which has been formatted for MATLAB, may be downloaded to accompany the case
study or application project. Solutions for the exercises are also available from the website. These resources have
been class tested and are an excellent source of out-of-class assignments.
CASE STUDIES
Chapter 1: Linear Models in Economics This case study examines Leontief’s “exchange model” and shows how
systems of linear equations can model an economy. Real economic data is used.
Chapter 4: Space Flight and Control Systems studies a mathematical model for engineering control systems. The
notion of rank is used to determine whether a system is controllable, and a system of equations is solved to
determine which inputs into the system would yield a desired output.
Chapter 5: Dynamical Systems and Spotted Owls examines how eigenvalues and eigenvectors can be used to
study the change in a population over time. Real data from populations of spotted owls, blue whales, and plants
(speckled alders) is studied, and the notion of a sustainable harvest is introduced.
APPLICATION PROJECTS
Section 1.2: Interpolating Polynomials shows how a system of linear equations may be used to fit a polynomial
through a set of data points. Polynomial curves are used to fit real data taken from Car and Driver magazine.
Section 1.2: Splines shows how a system of linear equations may be used to fit a piecewise-polynomial curve
through a set of data points. Cubic splines are used to fit real data taken from Car and Driver magazine.
Section 1.10: Diet Problems provides examples of vector equations that result from balancing nutrients in a diet.
between individuals and groups. The problem of ordering teams within a football conference is discussed, and real
data from various football conferences is used.
Section 2.1: Other Matrix Products introduces and explores the properties of two matrix products: the Jordan
product and the commutator product.
Section 2.3: Condition Numbers. This set of exercise motivates the definition of the condition number of a matrix,
Section 4.1: Hill Substitution Ciphers studies how matrices may be used to encode and decode messages. Matrix
arithmetic modulo 26 is used.
Section 4.6: Error-Detecting and Error-Correcting Codes studies how to construct methods for detecting and
correcting errors made in the transmission of encoded messages. The United States Postal Service bar code is
studied as an error-detecting code, and the error-correcting Hamming (7,4) code is also studied.
Section 7.2: Conic Sections and Quadric Surfaces shows how quadratic forms and the Principal Axes Theorem
may be used to classify conic sections and quadric surfaces.
Section 7.2: Extrema for Functions of Several Variables is designed for students who have experienced
multivariate calculus. Quadratic forms are used to investigate maximum and minimum values of functions of several
variables. Results are derived in terms of the eigenvalues of the Hessian matrix.
7. References
[1] D. Carlson, C.R. Johnson, D.C. Lay, A.D. Porter, “The Linear Algebra Curriculum Study Group
Recommendations for the First Course in Linear Algebra,” College Math. Journal (24), 1993, 41-46.
[2] D. Carlson, C.R. Johnson, D.C. Lay, A.D. Porter, A. Watkins, W. Watkins, eds., Resources for
Teaching Linear Algebra, MAA Notes, Mathematical Association of America, 1997.
[5] Marc E. Herniter, Programming in MATLAB
, Pacific Grove, CA: Brooks/Cole, 2001.
[6] Steven Leon, Eugene Herman and Richard Faulkenberry, eds., ATLAST Computer Exercises for
Linear Algebra, Englewood Cliffs: Prentice-Hall, 1996.