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COMPUTER PROJECTS 17
Chapter 4: Space Flight and Control Systems
This case study studies a mathematical model for engineering control systems. The central idea is
that the rank of a matrix determines if a system is controllable. A system of linear equations is solved
to determine which inputs into the system yield a desired output.
Chapter 5: Dynamical Systems and Spotted Owls
Chapter 7: The Singular Value Decomposition and Image Processing
This case study examines how the singular value decomposition of a matrix may be used to reduce
the amount of data needed to store a reasonable image of a graphical object. Two types of images
are considered: three-dimensional surfaces and black-and-white two-dimensional pictures.
F OVERVIEW OF APPLICATION PROJECTS
Section 1.2: Interpolating Polynomials
This set of exercises shows how to use a system of linear equations to fit a polynomial through a set
of data points. Polynomial curves are fit to actual acceleration data obtained from Car and Driver
magazine.
Section 1.2 Splines
This set of exercises shows how to use a system of linear equations to fit a piecewise-polynomial curve
through a set of data points. Cubic splines are fit to actual acceleration data obtained from Car and
Driver magazine.
Section 1.10 Diet Problems
18 COMPUTER PROJECTS
Section 1.10 Loop Currents
This set of exercises provides further and larger examples involving loop currents, and reinforces the
text’s development of this topic.
Section 2.1 Adjacency Matrices
Section 2.1 Dominance Matrices
This set of exercises utilizes matrices to address questions concerning competition between individuals
and groups. The problem of ordering teams within a football conference is discussed, and real data
from several collegiate and professional football conferences are used.
Section 2.1 Other Matrix Products
This set of exercises introduces and explores elementary properties of two matrix products: the
Jordan product and the commutator product.
Section 2.3 Condition Numbers
This set of exercises motivates the definition of the condition number of a matrix. It also explores
how the condition number affects the accuracy of solutions to a system of linear equations.
Section 2.6 The Leontief Input-Output Model
This set of exercises provides three real data examples of the Leontief Input-Output Model discussed
in the text. American economic data from the 1940’s and the 1990’s is studied.
Section 3.3 The Jacobian and Change of Variables
The Jacobian is derived and applied to the change of variables in double and triple integrals.
This set of exercises does make use of ideas from multivariate calculus.
Section 4.1 Hill Substitution Ciphers
COMPUTER PROJECTS 19
Section 4.6 Error-Detecting and Error-Correcting Codes
Section 5.3 The Fibonacci Sequence and Generalizations
This set of exercises introduces the Fibonacci sequence and Lucas sequences. Eigenvalues, eigen-
vectors and diagonalization are used to derive general formulas for an arbitrary element in these
sequences.
Section 5.4 Integration by Parts
This set of exercises shows how the matrix of a linear transformation relative to a cleverly chosen
basis may be used to find antiderivatives usually found using integration by parts.
This set of exercises does make use of ideas from integral calculus.
Section 6.4 The QR Method for Finding Eigenvalues
Section 7.2 Extrema for Functions of Several Variables
Quadratic forms are used to investigate relative maximum and minimum values of functions of several
variables. Results are derived in terms of the eigenvalues of the Hessian matrix.
This set of exercises does make use of ideas from multivariate calculus.
20 REFERENCES
G ADDITIONAL RESOURCES
There are many resources on the WWW that can be used in support of a course taught from this
book. Here is a short list of websites and other printed materials that that you might find useful in
support of your linear algebra course.
References
Books
[1] D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, “The Linear Algebra Curriculum Study
Group Recommendations for the First Course in Liner Algebra”, College Math. Journal (24),
1993, pp. 41–46.
Websites
[5] http://www.pearsonhighered.com/lay
The official online source for the Maple materials for the laylinalg package, each chapter’s
Case Study, the 21 Application Projects, and 16 Maple Projects. The Case Studies amplify
the vignettes that introduce each chapter, and the Application Projects cover a wide variety of
interesting topics. Many of these resources involve real world data, and they may be downloaded
as Maple worksheets. Sections 5 and E in this manual contain a synopsis of each Case Study
and Application Project. When you are planning your Maple assignments, you should review the
Case Studies and Application Projects. They have been class tested and are excellent out-of-class
assignments.
USING MAPLE FOR LINEAR ALGEBRA 21
6 USING MAPLE FOR LINEAR ALGEBRA
These instructions are written for Maple 14, but have been essentially unchanged since Maple 10.
Setting Up the Maple GUI
As Maple has matured new ways to interact with the software have expanded. The latest major
interface development appeared when Maple 10 was released in 2005. The new developments were
“documents” and “2D Math Notation.” Documents make it easier to create a professional-looking
document suitable for publication. 2D Math Notation means that mathematical expressions are type-
set using as much vertical and horizontal space as needed. Maple notation means that mathematical
expressions are displayed in a single line using Maple commands to represent all mathematical nota-
Figure 2: Snapshot of the Options popup window.
To tell Maple to start new windows as worksheets, select the Options item in the Tools menu
on the Maple window. A popup window will appear. Under the Interface tab, be sure the default
format for new worksheets is set to Worksheet. Within the Display tab the user can select the
specific way in which input, output, and graphics are displayed within a worksheet (or document).
22 USING MAPLE FOR LINEAR ALGEBRA
The final step is to confirm that your selections have been received. To do this, close the current
worksheet and open a new one. Assuming you followed all of the above recommendations, the new
document should be a worksheet in which input is entered using Maple notation, output appears in
2D Math Notation and graphics are displayed inline.
Linear Algebra in Maple
The principle way to define a matrix in Maple is with the Matrix command. (Remember that Maple is
case sensitive; maple is similar but different and should no longer be used.) The change from matrix
to Matrix occurred when the linalg package was replaced by the LinearAlgebra package. The
laylinalg package builds upon the commands in LinearAlgebra to create Maple implementations
of the MATLAB functions introduced in the Study Guide.
The following two commands should appear at the beginning of every worksheet that will utilize
the laylinalg package:
Palettes
Maple’s palettes can also be used to enter matrices (and vectors). Each palette contains a collection
of icons that, when clicked, simplifies the entering of mathematical expressions and reduces the need
to remember Maple syntax. For linear algebra, only the Matrix palette is directly appropriate. This
palette allows the user to specify the number of rows and the number of columns in the matrix. The
Type drop-down menu provides an easy way to generate a zero matrix, an identity matrix, a matrix
USING MAPLE FOR LINEAR ALGEBRA 23
Figure 3: Snapshot of a Maple window showing the matrix palette.
Labels
The Display tab of the Options window includes a check box labeled Show equation labels. When
this box is checked, every Maple output region receives a unique equation label very similar to what
you would find in a L
A
T
E
X document. These labels can be used in place of the more traditional history
operator (%).
To use a label in a Maple command, position the cursor where the label is to be inserted and
either expand the Insert menu and select Label or just type Ctrl–L. Type the equation label for
the equation you wish to use in the popup window that appears (the parentheses will be inserted by
Maple) and click OK. The label, surrounded by parentheses will be displayed at the current cursor
Context Menus
Acontext menu is a popup menu that contains the most likely operations to be performed on the
selected object. To activate a context menu, move the cursor to the object you want to manipulate
then press the right mouse button. Notice that the list of possible options includes assigning the object
to a name, browsing the matrix with Maple’s Matrix Browser (most effective for large matrices),
24 USING MAPLE FOR LINEAR ALGEBRA
Figure 4: A Maple worksheet showing the use of labels.
MAPLE PROJECTS 25
7 MAPLE PROJECTS
The following sixteen (16) projects are provided for you to use as is or to modify to fit the specific
needs and objectives of your course. Each project lists prerequisite sections of the text and the Maple
commands needed to complete the project. A summary of each project can be found in Section 5.D
of this document.
26 MAPLE PROJECTS
No mathematical information on this page.
This page left blank to facilitate duplex printing of individual projects.
Maple Project: Introduction to Maple NameMaple Project: Introduction to Maple NameMaple Project: Introduction to Maple Name
Purpose To begin to learn about Maple as a mathematical tool for linear
algebra and Maple’s online help system.
Prerequisites Elementary Algebra
Maple commands used help and ?
Notes:
•It often helps to do computer work with a partner. Help each other locate and fix typographical
errors, discuss Maple’s response, and ask and answer each other questions. If further experi-
ments are needed before answering the questions, feel free to do so.
1. (a) Start Maple, then for each of the following expressions, enter the expression in a separate
input region, execute it (by pressing the Enter key), and record Maple’s response to your
3 * x;
a*b^2;
(b) Explain the different results for a*b^2;and (a * b) ^2;.
2. (a) This set of commands will introduce you to Maple’s online help facility. There is a help
worksheet for every Maple command (including all commands in the LinearAlgebra and
laylinalg packages). Each help worksheet contains a full description of the commands
arguments and output and includes several examples to illustrate its usage. Read the
computer responses to the following instructions. You do not need to record the output.
If you want to take notes for your own purposes, use a separate sheet of paper. Do not be
concerned if a lot of the information in the help worksheets does not make sense to you
(yet).
i. help( LinearAlgebra );
ii. ?Matrix
Notes:
•When you are done with a Help window, click its close box in the upper right corner.
•The ?“command” is one of the few Maple commands that is not terminated with a
semicolon or colon.
•Type ?help for some additional information about Maple’s online help facility.
(b) Give a brief description of the Maple online help facility and how to access it.
(c) Explain how the Maple help system can be accessed via the Help pull-down menu.
Maple Project page 2 of 2 Introduction to Maple
Maple Project page 2 of 2 Introduction to Maple
Maple Project page 2 of 2 Introduction to Maple
Maple Project: Introduction to Linear Algebra with Maple NameMaple Project: Introduction to Linear Algebra with Maple NameMaple Project: Introduction to Linear Algebra with Maple Name
Purpose To learn about the basic linear algebra commands in the
LinearAlgebra and laylinalg packages.
Prerequisites Section 1.5
Maple commands used Angle brackets (< >), Matrix,Vector restart, and with.
Notes:
•To remove an assignment to a variable, say, x, and make xinto a variable again, use the
command: unassign( ’x’ );.
•Be sure you use the single quote (’), not the back quote (‘).
1. (a) For each of the following Maple commands, enter and execute the command, then record
the results in the boxes provided. Remember that each command must end with a semi-
colon or colon and that pressing the Enter key executes all commands in the current
execution group. (For more about execution groups and the worksheet interface, see
?worksheet.)
M;
N;
v := < 1, 2, 4, 8 >;
v;
Maple Project page 2 of 2 Introduction to Linear Algebra with Maple
Maple Project page 2 of 2 Introduction to Linear Algebra with Maple
Maple Project page 2 of 2 Introduction to Linear Algebra with Maple
Maple Project: Exchange Economy and Homogeneous Systems NameMaple Project: Exchange Economy and Homogeneous Systems NameMaple Project: Exchange Economy and Homogeneous Systems Name
Purpose Find the equilibrium price for an exchange model economy by solving
a homogeneous system.
Prerequisites Section 1.6
1. Let T =
.20 .17 .25 .20 .10
.25 .20 .10 .30 0
.05 .20 .10 .15 .10
,x=
x1
x2
x3
.
(a) Without using Maple, write out the five equations in this system.
(b) Collect terms in your equations to get a homogeneous linear system, and write out the
five new equations.
page 1 of 4
page 1 of 4
page 1 of 4
2. Let Bx=0denote the homogeneous system you obtained in Question 1(b), and calculate the
reduced echelon form of A = [B 0]. Record the reduced form in the table provided at the end
of this question.
The following Maple commands load the matrix T from the laylinalg package, create the
matrix B by subtracting the 5×5 identity matrix from T (see Section 2.1), and create the
augmented matrix A:
(a) Use bgauss,gauss, and scale to obtain the reduced echelon form of the augmented
matrix.
Notes:
•When you finish the forward elimination the (5,5) entry in the reduced matrix should
be very small. In fact, it is so small that you will not be surprised to learn that this
(b) The problems associated with floating-point arithmetic can be avoided by converting all
floating-point numbers in the original matrix to fractions (rational numbers). The simplest
way to do this is with the command:
A: original A: ref (floating point) A: ref (rational)
Maple Project page 2 of 4 Exchange Economy and Homogeneous Systems
Maple Project page 2 of 4 Exchange Economy and Homogeneous Systems
Maple Project page 2 of 4 Exchange Economy and Homogeneous Systems
3. Read about Leontief Economic Models in Section 1.6 of the text. Now consider an exchange
model economy which has five sectors: Chemicals, Metals, Fuels, Power, and Agriculture; and
assume the matrix T in Question 1 above gives an exchange table for this economy as follows:
C M F P A
.20 .17 .25 .20 .10
C
(a) Verify that each column of T sums to one. This indicates that all output of each sector
is distributed among the five sectors, as should be the case in an exchange economy. The
(b) Let xCrepresent the value of the output of Chemicals, xMthe value of the output of Metals,
etc. Using the reduced echelon form of [B 0] from Question 2, write the general solution
for Tx=x:
(c) Suppose that the economy described above is in equilibrium and xA= 100 million dollars.
Calculate the values of the outputs of the other sectors and record this particular solution
for the system Tx=x:
xC
xM
(d) Consider the matrices T and B created above. As previously observed, each column of T
sums to one. Consider how you obtained B from T and explain why each column of B
must sum to zero.
Maple Project page 3 of 4 Exchange Economy and Homogeneous Systems
Maple Project page 3 of 4 Exchange Economy and Homogeneous Systems
Maple Project page 3 of 4 Exchange Economy and Homogeneous Systems
Extra Credit
Let B be any matrix of any size with the property that each column of B sums to zero. Explain
Maple Project page 4 of 4 Exchange Economy and Homogeneous Systems
Maple Project page 4 of 4 Exchange Economy and Homogeneous Systems
Maple Project page 4 of 4 Exchange Economy and Homogeneous Systems
Maple Project: Rank and Linear Independence NameMaple Project: Rank and Linear Independence NameMaple Project: Rank and Linear Independence Name
Purpose To define rank and learn its connection with linear independence.
Definition: The rank of a matrix A is defined to be the number of pivot columns in A.
Example: Suppose the matrix A =
1 2 3 4
4 5 6 7
6 9 12 15
1 1 1 1
is defined in your Maple worksheet. When
1 0 −1−2
Notes:
•Recall that Dis a reserved name in Maple. To avoid problems, the matrix D is stored under
the name DD.
1. Use the two methods described above to find the rank of each of the following four matrices.
For each matrix, record the reduced echelon form, circle each pivot column, and record the rank
in the table on the next page. To get these matrices into your worksheet, use the laylinalg
command indat( );.
1 1 1 1 4 1
Matrix B C D, i.e., DD
Reduced
echelon form,
with pivot
columns
circled
Rank
Matrix E
Reduced
echelon form,
with pivot
columns
circled
Rank
2. Read the discussion of linear independence in Section 1.7. Write a definition for a linearly
independent set of vectors.
Maple Project page 2 of 4 Rank and Linear Independence
Maple Project page 2 of 4 Rank and Linear Independence
Maple Project page 2 of 4 Rank and Linear Independence
