INSTRUCTOR’S
TI CALCULATOR MANUAL
MICHAEL MILLER
Corban University
LUZ DEALBA
Drake 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.
Table of Contents
Foreword iv
Introduction iv
1. Getting Started 1
2. Planning the Course 2
Student Background
Planning Your Syllabus
3. Assignments and Projects 8
4. Guide to the Programs 11
5. References 18
6. Calculator Projects 19
Rules for Calculator Projects
Essentials for the TI Calculator
Systems of Linear Equations, Echelon Forms
Foreword
This manual is for any instructor who plans to use Texas Instruments calculators and Linear Algebra and
Its Applications together for the first time. It will greatly simplify your task of combining the calculator
with the text because it is written by a colleague who has successfully used the materials for several
years. 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 Michael Miller, who revised and expanded this TI
manual to include the TI-83+ and TI-89 calculators. This was a substantial undertaking, because each
Introduction
This Instructor’s TI-83+/84+(MP)/89 Manual will help you bring the handheld technology of TI
calculators into the classroom of your linear algebra course. Lay’s Linear Algebra and Its Applications is
a terrific text, and the calculator will help students get even more from the course.
In addition to general suggestions for incorporating the TI calculator into your course and using the
standard matrix menus, the manual describes the special programs, data, and projects that have been
developed specifically for use with Linear Algebra and Its Applications. The programs enrich the
capabilities of the calculators to automate routine tasks in ways that highlight key algorithms of linear
algebra. Data for over 900 text exercises can be downloaded into the calculator, which allows students
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1. Getting Started
Calculators, Equipment and Software
The Department of Mathematics at Corban University requires that students use a graphing calculator for
certain parts of every mathematics course. We strongly recommend a calculator from Texas
Instruments like one from the TI-83+ family (which includes the TI-84+), or the TI-89. The low
after class linking my calculator with students’ calculators to transfer the necessary information. The
programs, as well as data sets for problems from the text and for the projects, are available through the
Web site www.laylinalgebra.com that accompanies Linear Algebra and Its Applications, Fourth Edition.
Instructions for downloading and transferring the data are in a ReadMe file on the Web, specifically for
the TI Calculators.
transferring them to three or four students who, in turn, share with a few classmates, and so on.
Classrooms
An overhead projector is all you need for using a TI calculator in the classroom. The ViewScreen sets are
lightweight and it is no major problem to carry one to another building if necessary. You may want to
consider having your own ViewScreen set, then you can keep all of your work in your ViewScreen
2. Planning the Course
Student Background
At Corban, Linear Algebra has Calculus I as a requirement. For mathematics and computer science
majors, the first linear algebra course forms part of the core requirements.
Knowing something about your students will help you to tailor the course to their interests and abilities.
At the beginning of the course, I spend time making sure that students are acquainted with each other and
Planning Your Syllabus
Lay’s book contains a blend of topics that require analytical thinking and computations. This fact,
together with the large number of applications included in the text, allows for the linear algebra course to
be taught in a variety of ways. I have selected to use the core topics (except chapter seven) and have
Theory, Applications and Numerical Issues
For my classes, I have kept the level of the linear algebra course as high as possible. Although I do spend
most of the time discussing Rn, I also emphasize the importance of understanding definitions and
theorems, and how to use them. The fact that our linear algebra course is not dominated by mathematics
majors has not deterred me from presenting several formal proofs in class. Actually, I have found that the
introductory linear algebra course is the first course in which one can begin to demand from students a
certain level of abstraction. The spaces R2 and R3 can be studied through geometry, but as the dimension
Using the TI Calculator
The TI Calculator can be used for many of the situations that arise in the first linear algebra course. The
calculator is easy to use in general, and in particular, the matrix menus are manageable, so you may want
to explore in detail the regular use of a TI Calculator in this course. The most natural use of a calculator
in linear algebra would be to help with numerical computations, although with practice you can become
more sophisticated in the way you use the TI Calculator. I recommend that you use the calculator
frequently, or students will lose interest and you will find their papers saturated with computational
errors.
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I use calculators in the classroom in three different ways:
1) To introduce a feature of the TI Calculator that will simplify homework just ahead. I use
2) To explain a topic in linear algebra. For example, when teaching systems of equations I
use three good-sized matrices that represent systems of equations with zero, exactly one, and
3) To involve students in collaborative opportunities. Several times, during the semester, I
have my students work in groups of two or three, on a particular topic, during class time. For
example, I use worksheets, on determinants and matrix operations, which the students complete
as they read the book and calculate with their TI Calculator.
For simple arithmetic, my students are accustomed to the numeric format and data entry methods that the
TI-Calculator uses; the matrix menus also use familiar formats. As you and your students begin to feel
more comfortable with the use of the TI Calculator, you will want to use the calculator in your class more
often. There are a wide variety of topics for which you can design homework-type of calculator-based
activities; the choices that you make will depend on the focus of your course. Luz DeAlba suggests that
some of the pedagogical reasons for using calculator-based activities are that students can:
• Visualize mathematical concepts, like the action of linear transformations on the plane.
• Understand difficult topics through calculator projects.
By using a calculator in the course, I have also found that my students are more receptive to the reality
that mathematics does not revolve solely around computations, but also around the understanding of
Calculator Problems
Most of the calculator problems I have encountered arise because of round-off errors. I point out to my
students that extremely small entries (such as 1E–10) in a computation can often be taken to be zero. At
the beginning, most students expect the calculator to deliver the correct answer with 100% accuracy. It
takes a little time for them to learn that sometimes one can only get approximate answers to certain
questions.
The First Day of Classes
It is a good idea to mention to your students, the first day of classes, that you will be using the TI
Calculator in class; also, discuss the types of things that you will be doing with it, and then do them.
Encourage new users to work through “Quick Start,” the first chapter of the Guidebooks that came with
their calculators as soon as possible.
I definitely recommend that the TI Calculator be used immediately for row operations. If you do not use
it soon enough, you will find that students will not begin to appreciate the benefits of using a calculator
in linear algebra. Row operations are simple and students will think they can do them in their heads;
later, you will find your students’ work to be incomplete and plagued with errors. I recommend that,
during explanations in class, you yourself carry out the row operations with the calculator. That way the
students will see how efficient this can be and they will follow along. Also, I recommend that you give a
Study Guide
Lay’s Linear Algebra and its Applications, Fourth Edition comes with a supplement for students referred
to as the Study Guide; I encourage my students to use this Guide to help them in their study of linear
algebra. Lay has included “MATLAB boxes” throughout the Study Guide as he introduces technology;
these MATLAB boxes are gradually presented to the reader, explaining only what is necessary for the
section and homework exercises. I translated these boxes into notes in a format and language that is
specific for each model of TI Calculator. I tried to keep the emphasis and wording as close as possible to
Lay’s. These notes are included as an appendix to the fourth edition of the Study Guide.
Testing
For all testing I allow the students unlimited use of calculators. Of course this has changed the way in
which I construct my tests. Normally, I have six one-hour tests during the semester. On all tests I usually
have several short answer questions and a number of true/false questions that require short explanations;
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Sample Test Questions
The following problems are a small sample that I selected from the questions that have appeared in my
linear algebra tests since I adopted Lay’s book. Students are required to explain all of their answers. I
place great emphasis on writing and the use of correct terminology, as well as accurate mathematics.
1. Let A =
210
15 2
−
−−
⎡
⎣
⎢⎤
⎦
⎥
h
.
2. Let v1 =
1
1
0
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
, v2 =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2
3
1
, v3 =
1
2
4
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
. Determine if the vectors v1, v2 and v3 span R3. Based on your answer to
3. Let A =
21
13
10
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
and b =
1
4
1
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
. Assume that the vectors a1 , a2 are the columns of A. Answer the following
questions T (true) or F (false). Give a short explanation to support each answer.
T F The system Ax = b is consistent.
T F b is not in Span{a1 , a2}.
4. Let A =
122
252
382
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
.
6
5. Let A =
416
753
933
−
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
.
6. Let A =
50 1
132
05 3
−
−−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
. Answer the following questions.
7. Determine the dimension of W =
Span
-1
2
0
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
,,
0
1
1
1
3
1
. Write down a 3 × 4 matrix A such that
8. A linear transformation T : R4 → R3 has standard matrix representation given (this means that Tx = Ax). For b
9. A linear transformation T : Rn → Rm has standard matrix representation given. Answer the questions:
i. For each b in Rm, does there exist x in Rn such that T(x) = b? Explain.
ii. If the answer to i. is yes, how many vectors x are there in Rn such that T(x) = b? Explain.
⎣
⎦
⎣
⎦
10. Suppose that a 3 × 3 matrix has the characteristic equation below. Determine if the matrix is diagonalizable
and/or invertible, or explain why you need more information.
11. For each of the following matrices, state if the matrix is diagonalizable or not and give a short explanation to
support your answer.
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12. Let A =
100
84 6
−−
⎡
⎢
⎢
⎤
⎥
⎥
. Verify that v1 =
15
8
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
, v2 =
0
3
−
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
, and v3 =
0
2
−
⎡
⎢
⎢
⎤
⎥
⎥
, are eigenvectors of A.
13. Let A =
12 1
12 0 0
−
⎡
⎢
⎢
⎤
⎥
⎥
, and v =
h
2
⎡
⎢
⎢
⎤
⎥
⎥
.
14. Let W =
pq
pq
p
q
pq
+
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎫
⎬
⎪
⎪
⎭
⎪
⎪
3
: and real numbers
, be a subspace of R4.
15. For each one of the following cases give an example as indicated, or write down an argument to justify why
the example cannot be found. Also, write down a statement about the possible linear (in)dependence of the
columns of the matrices you created.
a. A 2 × 3 matrix P such that Px = 0 has a nontrivial solution.
3. Assignments and Projects
Preparing Your Own Assignments
You need time to prepare your own assignments, and these need to be carefully planned. You may
conceive a project that you think is useful for your class, but by the time you type it and perfect it, you
may find that the moment for the project is not appropriate. Too much time might also have passed since
you covered the topic in class. For first time calculator users, I suggest that if you create your own
Grading the Assignments
Included in this manual are some of the calculator assignments Luz DeAlba has used over the past few
years. In addition to computations, all solutions require explanations in good English with complete
sentences. She developed a grading system that uses 12 points and plus-minus grades A+ = 12 to F = 0.
She suggests reviewing each project twice. The first time through, she reads the projects thoroughly
Comments About the Calculator Projects
Over the next few pages, you will find descriptions of the various calculator projects listed in this
manual. Included in the descriptions are lists of programs that will be used as well as an account of the
specific sections of the text that each project covers. The PROJDATA program contains appropriate data for
each project.
Essentials for the TI Calculator
This handout contains explanations for the various TI Calculators (TI-83+, TI-84+(MP), TI-89) on using
Project 1: Systems of Linear Equations, Echelon Forms
This project needs to be done early in the term, so that the students will learn the value of using a
Project 2: Span and Linear Independence
This project uses the programs GAUSS and BGAUSS, although it can be done by using only row operations,
I use technology in this project with the purpose of eliminating computational difficulty so that the
student can concentrate on the concepts of Span and Linear Independence. The material in Sections 1.3,
1.4, 1.5, and 1.7 of the text is necessary.
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Project 3: Linear Transformations
The first problem of this project helps the student to visualize the action of linear transformations on the
plane. The second problem is a good project in which the properties of linear transformations are
essential, although some students are very good at guessing the answers. Both problems require the use
Project 4: Matrix Algebra
This project is designed to be completed in class, and I do not lecture on this topic. It is the students’
responsibility to read the textbook and finish the project at home if they were absent or fell behind in
their work during class time. I break up my class into groups of two or three—keeping students with the
Project 5: LU Factorization
This project uses the programs GAUSS and BGAUSS. I use technology in this project with the purpose of
eliminating computational difficulty. I usually assign this project as extra-credit. The material in Section
2.5 of the text is necessary. A Web application project for Chapter 2 uses the LU factorization to
construct a QR factorization (which is introduced in Exercise 24 of Section 2.5 in the text).
Project 6: Leontief Input-Output Model
This project is based on Exercise 14 of Section 2.6. The use of technology in this project is essential
because of the nature and size of the data used. I usually assign this project as extra-credit. The material
Project 7: Determinant and its Properties
This project is designed to be completed in class, and I handle it in the same manner as Project 4. This
project can be done just before starting Section 3.2, so students can discover properties of determinants
Project 8: Cryptography
The purpose of this project is to give students another opportunity to study an application of linear
algebra, in particular row operations and the inverse of a matrix, to a real world problem. The material in
Sections 2.2, 3.1, 3.2, and 3.3 is necessary. Data for this project are on the Web described in Section 1.
The Web contains a more elaborate investigation of cryptography using modular arithmetic. See the
Chapter 4 application project on Hill substitution ciphers.
Project 9: Intersection of Two Vector Spaces
The purpose of this project is to give students an opportunity to work on a more abstract problem. The
use of technology in this project reduces errors. I usually assign this project as extra-credit. This project
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Project 10: Null Space and Column Space
This project was designed with the purpose of reinforcing the materials in Sections 4.1 – 4.6 of the text.
Technology is used to determine if certain systems of equations are consistent and for finding solutions.
This is the first opportunity that the students have of using the row reducing operations ref( and rref(
Project 11: Dynamical Systems
This project extends the predator prey model of Section 5.6, using ideas from Sections 5.1, 5.2 and 5.6.
The Chapter 5 Case Study on the Web extends the stage-matrix model for the spotted owls, and it
includes real data for blue whales and plants.
Project 12: Gram-Schmidt Process and QR Factorization
This project uses the programs PROJ and PROJV. I use technology in this project with the purpose of
eliminating computational difficulty. Problem 1 requires a geometrical interpretation of the results. The
Project 13: Least-Squares Approximation
The purpose of this project is to give students another opportunity to study an application of linear
algebra to a real world problem; you should feel free to use the ideas in this project with your own data. I
use technology in this project with the purpose of eliminating computational difficulty and for graphing.
The material in Sections 6.5 – 6.6 of the text is necessary. The Chapter 6 Case Study on the Web has
interesting data sets (including sinusoidal data) that could be added to this project. The regression
capabilities of the TI Calculators can be used to compare the results obtained by linear algebra methods.
The table below shows at what point in the course these projects can be presented to the class:
Project 1 Section 1.5
Project 2 Section 1.6
Project 3 Section 1.9
Project 4 Section 2.1
4. Guide to the Programs
Program Information
The programs below can be downloaded to your computer from the Web site www.laylinalgebra.com.
These programs have proved helpful in simplifying the work, but it is always important to know and
understand the theory behind the calculations. To download these programs from the Web site and into
your TI Calculator you need the equipment described in Section 1. The web site contains a ReadMe file
TI-83+ and TI-84+ (MP)
The 8 button displays a menu containing the names of the programs that are stored in the calculator’s
memory (in alphabetical order); you can scroll down using the down arrow to see all of the programs.
The programs that were written for this text are all embedded in the program called ALINEAR. Press ¸
on the ALINEAR program, and use the arrows to scroll to the desired program. Press ¸ when the
Figure 1: Program menu Figure 2: Program prompt
The output of a program is automatically displayed on the output screen, where scrolling is not possible. I
designed the programs so that the output is also stored in the system variable Ans. This variable is useful
TI-89
From the home screen, press the 2 ° buttons display a menu containing the names of the
programs that are stored in the calculator’s memory (in alphabetical order); you can scroll down using
the down arrow to see all of the programs. The programs that were written for this text are all embedded
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available. Use the up and down arrows to scroll and press ¸ on the desired program to begin
execution of the program. (See Figure 4.) All the programs described below will prompt the user for
Figure 3: Program start-up Figure 4: Program menu Figure 5. Program prompt
The output of a program is automatically displayed on the program output screen, where scrolling is not
always possible. To perform other functions, you must press the “ key to return to the home screen.
Each one of the programs below is described in terms of input needed and output created. In most of
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Program Descriptions
BGAUSS Clears entries in a column of a given matrix above the pivot in a specified row.
INPUT: A matrix, a row index.
OUTPUT: An error message if there is no pivot in the specified row; a matrix with zeros
above the pivot entry in the specified row, otherwise.
TI-83+ and TI-84+ (MP)
Usually, you will use this program several times (along with *row(, in the
2 > MATH menu) to reduce a matrix in echelon form to one in reduced
TI-89
Usually, you will use this program several times (along with mRow(, in the 2
I 4:Matrix J:Row ops menu) to reduce a matrix in echelon form to one in
reduced echelon form. The output of this program is saved in the system variable
CHARA Computes the coefficients of the characteristic polynomial of a square matrix.
The program is limited to matrices of size up to 3 × 3.
INPUT: A matrix.
OUTPUT: A column vector containing the coefficients of the characteristic polynomial of
the matrix, in descending order of the powers of x.
CNORM Computes the largest of the sums of the absolute values of the elements in each
column. (TI-83+ family only)
INPUT: A matrix.
OUTPUT: The largest sum of the absolute values of the elements in each column.
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GAUSS Clears entries in a column of a given matrix below the pivot in a specified row.
TI-83+ and TI-84+ (MP)
Usually, you will use this program several times (in conjunction with rowswap(,
in the 2 > MATH menu) until a given matrix is reduced to a desired echelon
form. The output of this program is saved in the system variable Ans, as well as
in a matrix [J]. You can recall the variable Ans by pressing 2 Z. This
feature is useful in repeated applications of the program; when the matrix is not
totally reduced, and no row exchanges are required. If row exchanges are
needed, swap the rows of [J] and then use [J] as the input.
JCOL Extracts a specified column of a matrix.
INPUT: A matrix; a column index.
OUTPUT: A column vector containing as entries the indicated column of the given matrix.
JROW Extracts a specified row of a matrix.
INPUT: A matrix; a row index.
OUTPUT: A row vector containing as entries the indicated row of the given matrix.
LU Computes most of the LU factorization of a matrix. (TI-83+ family only)
OUTPUT: A matrix whose columns form a basis for the null space of the input matrix; if
the columns of the input matrix are linearly independent then it returns {0}. The
set of vectors obtained using this program is the same set you get, if hand
calculations are made, based on the reduced echelon form of the input matrix.
PRDCT Computes the product of the entries in a column vector.
RCOL Replaces a specified column of a matrix with a given vector.
INPUT: A matrix, a column vector, and a column index.
OUTPUT: A matrix obtained from the input matrix by replacing the indicated column with
the input column vector.
RDIAG Computes a random diagonal matrix.
the interval [–9, 9].
UNITV Returns a column vector of length one in the direction of v.
INPUT: A column vector.
OUTPUT: A column vector whose length is one, and is in the same direction as the input
vector.
TI-89