Education Maple Manual Homework Label Reference Repeating This Process Unchecks The

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: 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: 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

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

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

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: 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