INSTRUCTOR’S
MAPLE MANUAL
DOUGLAS MEADE
University of South Carolina
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.
4 PURPOSES FOR COMPUTER EXERCISES 6
A POSSIBLE GOALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
B YOUR GOALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
C COMPUTATIONAL WISDOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
D OTHER POSSIBILITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7 MAPLE PROJECTS 25
Introduction to Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 pages
Introduction to Linear Algebra with Maple . . . . . . . . . . . . . . . . . . . . . . . . 2 pages
Exchange Economy and Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . 4 pages
Rank and Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 pages
The Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 pages
Pseudo-Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 pages
iv
Introduction
This Instructor’s Maple Manual provides a wide variety of information, support, and supplemental
materials for an instructor teaching a first course in linear algebra from Linear Algebra and Its
Applications who is interested in enhancing the course by incorporating the use of the Maple computer
algebra system.
In addition to general suggestions for incorporating Maple into your course, the manual includes
sixteen projects. The projects tend to be shorter than the Application Projects and Case Studies that
are on the text’s website. Each project contained in this Instructor’s Manual may be downloaded as
a PDF file from the Web. Each Application Project and Case Study is available for download in a
variety of formats, including as Maple worksheets. Since many students may have no prior exposure
to Maple, most of the project descriptions include the Maple code needed to complete a project. The
goal of the projects is to teach linear algebra, not Maple. However, Maple skills acquired here will
be directly applicable to other areas of mathematics, science, and engineering.
While the broad selection of projects is very appealing, the textbook remains the cornerstone of
the course. The exercises are carefully selected to complement and supplement the exposition. The
matrices and vectors — numeric or symbolic — for nearly 1000 exercises can be accessed in Maple
with a few keystrokes. Also, special laylinalg commands implement the matrix operations exactly
as they are described in the text. This allows students using Maple for their homework to focus on
the mathematics without worrying (as much) about correctly copying the problem from the book or
making algebra errors. When my students realize these benefits, their use of Maple on homework
skyrockets.
GETTING STARTED 1
1 GETTING STARTED
PREPARE
The number of options for using technology in support of a course in linear algebra continues to grow.
Commercial software packages MATLAB, Maple, Mathematica, and MathCad, and calculators such
as the TI-86, TI-92, and HP-48G can all be used together with Linear Algebra and Its Applications,
Fourth Edition. The [M] exercises in this text are written in a software-independent style and can be
solved with any of these tools. This manual, however, is written specifically for instructors interested
in using Maple.
This manual also assumes that you have the Study Guide for Linear Algebra and Its Applications,
Fourth Edition and the laylinalg package, a collection of Maple procedures and data for nearly
1000 exercises in the text. The Study Guide contains a self-contained guide to using technology with
this text. Although the main sections of the Study Guide discuss MATLAB, there are appendices
for Maple, Mathematica, and several graphing calculators. The Maple appendix provides explicit
STUDENT COPIES OF MAPLE AND THE STUDY GUIDE
Students should have a personal copy of the Study Guide. Maplesoft has a number of programs for
providing a copy of Maple to students at quite reasonable prices. Some campus site license agreements
might include free copies of Maple for your students. If not, instructors can often arrange better prices
for their students if they register with the Maple Adoption Program (http://www.maplesoft.com/
academic/adoption/). You should consult with Maplesoft and your Addison-Wesley representatives
to determine the best option for your students.
THE laylinalg PACKAGE
The laylinalg package is a Maple library that contains additional Maple commands and data for
almost 1000 exercises in the text (both regular and [M] exercises). This package and installation
2GETTING STARTED
Guide. It can also be effective to demonstrate the use of these tools as a part of your lectures. A
complete list of Maple commands provided in the laylinalg package can be found in Section 3.A.
The individual commands are described in more detail in the Study Guide.
THINGS TO ANNOUNCE AND TO FIND OUT
At the first class inform the students that, if they plan to use Maple outside your labs, they will
need a copy of the laylinalg package. It is anticipated that most students will download the
laylinalg package through http://www.pearsonhighered.com/lay. However, depending upon
your site license and you local IT infrastructure, you might want to have them copy laylinalg to a
portable storage device. (Make a ZIP archive first and have the students copy just one file.)
It is a good idea to distribute a survey at your first class meeting to find out your students’
majors and the previous experience they have with computers, calculators, and various types of
software. This will give you an idea of your students’ background (with regards to mathematics and
ASSIGNMENTS
The first time you plan to assign computer exercises, you might want to proceed with some caution.
Keep a close watch on students’ difficulties with each assignment. Various topics can require more
attention than you might expect, especially at the beginning of the course. For instance, you may
find that access to the computer labs is inadequate, maintenance is not as quick as you would like,
or students who buy the software have trouble installing it. Equipment has a knack for picking the
worst and most unlikely times to act up. Assume “If it can go wrong, it will.” and then be pleasantly
surprised if things go smoothly.
Your first assignment should not be about math. Your students will panic when something does
not work perfectly for them. Have them practice using the computers, saving and transporting
their work, and then submitting the assignment in the form you choose — printed and handed in,
submitted to your local course management system, or sent via e-mail. If you would like them to
do some math for their first assignment, do not grade it based on the solutions. You might use the
opportunity to send back comments on how you would like solutions presented. Then talk about how
computer skills are important. Collect and grade both kinds of problems.
If you are able, find a student grader who has experience working with a computer algebra system
(CAS) — preferably Maple. He or she can handle a large portion of the routine problems, leaving
you time to focus your efforts on more conceptual exercises. Having an assistant in the lab during
some hours is also a huge benefit, especially during the first weeks of class.
ALLOW TIME TO ADAPT, TO THINK, AND TO BE CREATIVE
It would be very good to have some release time the first time you incorporate the use of technology
in a course. Most likely, you will find yourself spending more and more time creating the kinds of
exercises you wish you could have assigned in the past. Maple can provide you with new opportunities
to be creative. But, be realistic as making changes in a course always takes more effort than expected.
Your students will have different interests and questions. Do not be afraid to modify the pace, style,
or even topics of the course as it proceeds.
Along the same lines, it is recommended that you teach the course more than once. This pro-
vides opportunities to improve your materials and methods and increases the return on your initial
preparation for the course.
The very existence of powerful and accessible matrix computation tools raises questions about
what topics to emphasize, what skills are most important, and what style of teaching is best. These
COMMON DIFFICULTIES AND OPPORTUNITIES
Many of your students will be very computer literate, but there may be a few with no prior computer
experience. The Maple interface and mechanics will be alien to these students. Editing, saving,
printing, and executing will be new skills. These skills are not pertinent to linear algebra, but they
are central to operating the software. You can avoid such problems by having all results written by
hand and rarely printing graphics. However, this keeps many students in the dark and allows anxiety
4THE COMPUTING ENVIRONMENT
would like homework handed in, but do not restrict the students to a minimal set of skills. Answer all
of their questions and solicit help from the rest of the students. The class runs much more smoothly
when everyone feels free to experiment and to ask questions.
THINGS KEEP CHANGING
At some point, a student may report different numerical results than you anticipate. Do not worry.
Different floating-point processors carry different numbers of significant digits. Most CAS will let
you change this setting. View these episodes as opportunities to discuss what is going on. After all,
this is the type of problem your students will run into frequently when they are on the job. It is not
2 THE COMPUTING ENVIRONMENT
A COMPUTING LABS
Your department may have several public labs equipped with PCs. They could be networked including
a printer somewhere. Depending on the setup and your license agreement, you may want to install
Maple on each individual machine or on a server. If students are required to have an account to access
e-mail and the Internet, you probably will want to locate the system administrators and inform them
of your plans. They can help you a lot, but usually request enough advance notice to investigate,
test, and implement the necessary installation for you and your students.
B HOW STUDENTS DO THEIR COMPUTING
The simplest setup would be for everyone to use the same hardware and to be in the lab at the same
time. Get that picture out of your head. Some students will use the lab. An increasing number of
students will have computers at home and will want to work there also. When they do use the lab,
their attendance will be scattered throughout the day. Some students may find other campus labs
CLASSROOM DEMONSTRATIONS 5
3 CLASSROOM DEMONSTRATIONS
A SOFTWARE
In addition to the laylinalg package, the built-in LinearAlgebra,plots, and DEtools pack-
ages are the only other Maple libraries needed for this course. While there appears to be some
duplication between the functionality of commands in the LinearAlgebra and laylinalg pack-
ages, e.g., GaussianElimination and gauss, the laylinalg commands have been designed so that
Table 1: Brief description of the commands defined in the laylinalg package.
Function Description
bezier Plots the control points and associated B´ezier curve (degrees 2, 3, & 4)
bgauss (Backward Gauss) Uses the leftmost nonzero entry in a selected row as
the pivot, and creates zeros in the pivot column above the pivot entry
gauss Uses the leftmost nonzero entry in a selected row as the pivot, and uses
row replacement to create zeros in the pivot column, below the pivot
entry or in specified rows
gs Performs the Gram-Schmidt process on the columns of a matrix
6PURPOSES FOR COMPUTER EXERCISES
loading the laylinalg package then issuing the Maple command: c1s1( 25 );. Electronic access to
the exercise data frees the student from the burden of manually entering (and checking) the matrices
and vectors required to solve a problem. It can also be a timesaver for instructors.
A Maple session can be saved as a worksheet. Maple worksheets are platform independent. That
is, regardless of the computer on which the worksheet was created (Windows, Macintosh, Linux,
B EQUIPMENT
For demonstrations to be readable by a classroom of students some way to enlarge and project
the computer screen is needed. The most convenient arrangement is a projection unit hooked to
a computer. It is increasingly common to be able to connect your laptop directly to a projector
permanently mounted in a classroom. But, sometimes it is still necessary to make arrangements to
bring a portable projector into the classroom.
4 PURPOSES FOR COMPUTER EXERCISES
A POSSIBLE GOALS
As you consider your students’ interests and begin to appreciate the potential of computer exercises,
decide what goals are most appropriate for your class.
Here are some possibilities to get you started:
•To teach applications
•To reinforce understanding of concepts
•To think and solve
PURPOSES FOR COMPUTER EXERCISES 7
B YOUR GOALS
Your goals may or may not intersect with the list above. Technology can open many new doors
simultaneously; it is not a bad idea to sit down and identify the one or two that are most important
to you. Do not shoot for too many at once. I believe it is much better to accomplish a few goals
C COMPUTATIONAL WISDOM
Computational wisdom refers to the fact that your students are likely to be doing complex calculations
when they enter the workforce. They need experience with real-world data and this may be one of the
few classes able to introduce them to some of the potential pitfalls. There is no reason to form each
exercise to demonstrate possible trouble, but an honest approach to the situation will be appreciated.
You might wish to emphasize things such as:
D OTHER POSSIBILITIES
David Lay and I both believe that both applications and theory are very important. He covers more
of the text in lectures and makes fewer explicit computer assignments than I do. You will have to
decide how much material you would like the students to learn from their computer work. It is all
a matter of taste and style; you are encouraged to develop a course that is appropriate for you and
your students. (See also [1] and [2].)
E SAMPLE EXAM QUESTIONS
One way to emphasize the importance of the computational or graphical work done in the projects
is to include questions on the exam that are based on the results obtained in the laboratory. If you
permit students to complete the projects in groups, such questions are also effective for emphasizing
that all members of the group must be active participants in the project.
Here are some sample questions seen on linear algebra tests where computers were used during
instruction — but not during the test.
8PURPOSES FOR COMPUTER EXERCISES
2. Let A =
1−3−2
2−1 3
−1 4 −5
and b=
10
−22
32
.
(a) Does the system Ax=bhave a solution? Is the solution unique? Explain your answers.
(b) Is bin the span of the columns of A?
(c) Do the columns of A span R3? Explain your answer.
4. Let A be a 2 ×3 matrix.
(a) If the system Ax=bis consistent, is the solution unique? Why or why not?
(b) Suppose A has 1 pivot column. Is the system Ax=bconsistent for any choice of bin
5. There is a real 3×3 matrix A for which the general solution to Ax=
1
−2
3
is x=c
−3
4
1
+
1
−2
. What is the general solution to Ax=0?
6. If the columns of an n×nmatrix A are linearly independent, does A−1exist? If this inverse
exists, are its columns linearly independent? Explain.
10. Is it possible that all solutions of a homogeneous system of six linear equations in eight unknowns
are linear combinations of two fixed non-zero solutions? Explain.
11. A certain population of owls feeds almost exclusively on wood rats. The following matrix
describes the evolution of the owl and rat populations from one year to the next:
(a) Suppose you want to find the number of owls and rats five years from now. What would
you calculate, why would it work, and how would you interpret the results to provide the
populations five years in the future?
12. If the stock market went up today, historical data shows that tomorrow it has a 65% chance of
going up, a 10% chance of staying unchanged, and a 25% chance of going down. If the market
is unchanged today, tomorrow it has a 20% chance of being unchanged, a 40% chance of going
up, and a 40% chance of going down. If the market goes down today, tomorrow it has a 25%
chance of going up, a 10% chance of being unchanged, and a 65% chance of going down.
(a) If the market went up today, what is the probability that the market is unchanged the
day after tomorrow?
13. We stored a certain 3 ×3 matrix A in Maple, did several row operations to A, and ended up
with
14. Let A be a square matrix. Give an example to show that A and 2A do not usually have the
same eigenvalues. Are the eigenvectors of A and 2A always the same? Explain.
15. Let A be the n×nmatrix in which every entry is 1. Justify your answers to the following
statements and questions.
16. Suppose A is a 7 ×7 matrix with four distinct eigenvalues.
(a) Is A diagonalizable? Why or why not? (Do you have enough information to answer this
question?)
17. Let A = “16 −4
−4 1 #and V = “1−4
4 1 #. Verify that the columns of V are eigenvectors for A.
18. Consider A =
011
and b=
5
.
19. Suppose that two partitioned matrices satisfy “I X
0Y#” I B
0D#=“I0
0I#, where X,Y,
B,D, and the identity matrix (I) are all n×n. Find formulas for Xand Yin terms of Band
D.
PURPOSES FOR COMPUTER EXERCISES 11
20. The diary of a Maple session with most Maple commands omitted is reprinted below. Fill in
each blank with the appropriate Maple command that produces the listed output.
> restart;
> with( laylinalg ):
>
> M1 := < < 0, 3, -6, 6, 4, -5 >,
> < 3, -7, 8, -5, 8, 9 >,
> < 3, -9, 12, -9, 6, 15 > >;
[0 3 -6 6 4 -5]
[ ]
> M2 := __________________________________ ;
[3 -9 12 -9 6 15]
> M3 := __________________________________ ;
[3 -9 12 -9 6 15]
[ ]
M3 := [0 2 -4 4 2 -6]
[ ]
[0 3 -6 6 4 -5]
> M4 := __________________________________ ;
[3 -9 12 -9 6 15]
[ ]
12 COMPUTER PROJECTS
5 COMPUTER PROJECTS
A GENERAL INFORMATION
The projects in this manual are based on material in Linear Algebra and Its Applications, Fourth
Edition, and on contributions from various workshops. They enrich and expand the text material and
are independent of each other. They can be used as assignments or as extra credit. You may copy
and use them as written, or adapt them to suit your own situation or interests. The time required
B PARTNERS
Consider having your students find partners for the computer work. After a couple of days, you
may have to step in and assign partners for those who have not found partners yet. The computer
projects generally go much more smoothly when students work together. Have the partners turn in
one project with both names. This will reduce the workload on you, as well as give the students
experience working with others. Tell them not to hesitate to inform you if their partner is not keeping
C GRADING
You will have to decide how much weight to assign for computer projects. A good starting point is
about double that of a homework assignment that requires some writing. Consider more weight for
longer projects. If you are assigning a computer project once a week, then this will also need to be
Instructions for Students
•You are expected to use Maple to complete this project. The general objective of this project is
to explore topics discussed in class, to deepen your understanding of the computations involved
in linear algebra, and to gain an appreciation for the diverse applications of linear algebra. In
some cases you will be expected to read some background materials to learn about topics not
directly discussed in class. You will also develop your mathematical writing skills.
•The due date for the project will be announced. Points will be deducted for late work. Work
turned in one day late will receive 0 points in the “on time” row of the grading sheet, work
Grading Sheet for Maple Projects
Name:
Points Points
Possible Grading Criterion Awarded
11 Correct mathematics
3 Appropriate mathematical notation
2 Prose is clearly written
1 Prose is included appropriately
Figure 1: Sample cover page for student projects includes some general comments and guidelines and
a scoring rubric that shows the point distribution. Note that these instructions allow for group work
but require individual final reports.
14 COMPUTER PROJECTS
D OVERVIEW OF MAPLE PROJECTS
Project 1 — Introduction to Maple
This project is intended for students with no prior Maple experience. The purpose of this project
is to help students become comfortable with the Maple worksheet interface, online help, and the
laylinalg package. The mathematical content is minimal.
Project 2 — Introduction to Linear Algebra with Maple
Project 3 — Exchange Economy and Homogeneous Systems
This project is self-contained, but is related to Example 1 in Section 1.6. Students can read the
background information on their own. Note the discussion of floating-point and exact arithmetic.
Students need to be aware of the potential problems that can arise with floating-point arithmetic.
This issue deserves some attention in class – but not too much.
Project 4 — Rank and Linear Independence
A computational definition of the rank of a matrix is provided at the beginning of this project.
Otherwise, this project requires only Section 1.7. Students practice applying the definitions. The
exploration of linear independence can facilitate a better understanding of this fundamental concept.
Project 5 — Population Migration
Students like linear dynamical systems and plotting. This project is essentially Exercise 10 (page 87)
in Section 1.10 with some graphical additions. Use Example 3 (page 85) in Section 1.10 to introduce
the project. If you want to go further, ask the students what they expect to happen if this pattern
of migration continues indefinitely. Does everyone move to the suburbs? (Why?)
Project 6 — Initial Analysis of the Spotted Owl
This is very similar to the Population Migration project. Students will need to read the Case Study
Project 7 — The Adjacency Matrix of a Graph
This project is a detailed examination of matrix multiplication (Section 2.1) and its applicability in
graph theory. While the mathematics is simple, the application is interesting.
Question 8 asks students to create a definition. This type of question will be new for most
students, but I believe it is something they should experience. A common first response is: “A
dangerous worker is any worker with the highest level one contact.” A difficulty with this definition
is that contact level relates two workers. An improved definition is “A dangerous worker is any worker
Project 8 — An Economy with an Open Sector
This project begins by solving Exercise 13 (page 137) in Section 2.6. It then asks students to look at
the structure of the problem in more detail. Question 3 is designed to test students’ understanding
of Theorem 11 (Section 2.6, page 134). This connection can be provided as a hint, but resist the urge
to give away too much information. (See also Case Study 1: Linear Models in Economics.)
Project 9 — Curve Fitting
This project describes how linear algebra can be used to find a polynomial that interpolates a col-
lection of data. Vandermonde matrices are mentioned, but only to guarantee that the system is
invertible. (Vandermonde matrices appear in Exercise 11 (page 160) in the Supplementary Exercises
Project 10 — Temperature Distributions
Students use a characterization of the steady-state temperature (average of the temperatures at
the four neighboring nodes) to see how steady-state temperatures can be computed using linear
algebra. Other temperature distribution problems in the text can be found in Exercises 33 and 34
for Section 1.1 and Exercise 31 for Section 2.5. More examples are easily created. (See also the
Application Project for Section 2.5: Equilibrium Temperature Distributions.)
Project 11 — Manipulating Matrices with Maple
The project introduces the family of square matrices Mn= [max(i, j)]. Based on selected (small)
examples, students are asked to formulate conjectures for the formulas for the determinant (Sec-
tion 3.2) and inverse (Section 2.2) of Mnfor all positive integers n. This is much simpler than it
sounds; students have a lot of fun with this one.
Project 12 — Markov Chains and Long-Range Predictions
Markov chains are introduced in Section 4.9, but this project can be assigned after completing
Section 2.1. Only terminology is needed from Section 4.9. With appropriate guidance, this project
16 COMPUTER PROJECTS
Project 13 — Real and Complex Eigenvalues
This project can be a substitute for a formal discussion of Section 5.5. In particular, this project
Project 14 — Eigenvalue Analysis of the Spotted Owl
This is the continuation of the Maple Project 6: Initial Analysis of the Spotted Owl. The first three
questions are fairly self-explanatory. Eigenvalues and diagonalization are emphasized. The plots help
bring everything together. The extra credit is much more involved. The symbolic capabilities of
Project 15 — The Cayley-Hamilton Theorem
The Cayley-Hamilton Theorem is stated. The students are asked to verify the theorem for randomly-
selected matrices of various sizes. The reference for this project is Exercises 5–7 in the Supplementary
Exercises for Chapter 5). This project can be assigned anytime after Section 5.2 has been discussed.
Project 16 — Pseudo-Inverse of a Matrix
This project can be used as an application of the Invertible Matrix Theorem and as an introduction
to the Moore-Penrose inverse (in Section 7.4). The pseudo-inverse is shown to be theoretically
useful in least-squares problems (as presented in Section 6.5). This is demonstrated numerically and
graphically for a specific example.
E OVERVIEW OF 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 2: Computer Graphics in Automotive Design
This case study explores the effective two-dimensional rendering of a three-dimensional image. Per-
spective projections, rotations, and zooming are discussed and applied to wireframe data derived
from a 1983 Toyota Corolla.