5. References
[1] D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, Ann E. Watkins, and William Watkins, eds.,
41-46.
[3] K. Donnelly, MATLAB® Manual: Computer Laboratory Exercises, Saunders, 1995.
[4] D. A. Hill, D. R. Zitarelli, Linear Algebra Labs with MATLAB®, Macmillan, 1994.
18
6. Calculator Projects
This section contains some calculator projects that have been used in Luz DeAlba’s courses and in mine
as well. Feel free to modify these projects and adapt them to your course. Also included are the rules for
the projects, as well as the rubric developed by Professor DeAlba. You may want to visit the Web site
www.laylinalgebra.com for additional updates on the topics in this section.
• Any computation done with a calculator should contain explanations (written in good English) and
should indicate that you are able to associate the calculator work to the problem being solved and to
the materials presented in the classroom.
• If you use your calculator to graph, you must copy the graphs onto a sheet of paper and add a written
statement showing that you are able to associate the graphs to the problem being solved and to the
• draws diagrams (if appropriate) to support explanations.
11 points – Very Good
• shows thorough understanding of the material,
• computes correctly and checks solutions,
• uses flawless logic and draws valid conclusions from hypotheses,
• may have minor errors in the use of mathematical terminology,
• uses calculator correctly and efficiently,
• draws diagrams (if appropriate) to support your explanations.
5 points – Lacking
• shows lack of understanding of major components of the material,
not,
• writes weak or incorrect explanations in fragmented sentences,
• exhibits major errors in the use of mathematical terminology,
• uses calculator all the time, even when simple arithmetic is used.
0 points – Does not belong in this course
keys. The sequence 2 1 sets the ALPHA-lock.
§ Stores answers, matrices, quantities, etc. into variable names
2 [ The calculator will display your last entry. To recall the last entry
you press 2[, to recall the second to last entry press: 2 [
2 [, and so on.
TI-83+
Entering Matrices: There are several ways to enter matrices in the TI–83+. One method consists of
entering the matrix directly as an expression, while another method is through the use of the matrix editor
Direct Entry: Begin a matrix with a square bracket, and each row of the matrix with a matching pair of
square brackets; separate the entries in the rows with commas. Figure 1 below shows how to enter the
matrix
123
456
⎡
⎣
⎢⎤
⎦
⎥
directly and assign it the name [A] from the 2 > NAMES menu. The symbol !
appears after pressing the § key.
Essentials for the TI-83+ and TI-84+ (MP) 2
TI-84+ (MP)
Entering Matrices: There are several ways to enter matrices into the TI-84+(MP). The methods
described earlier for the TI-83+ will also work for the TI-84+(MP). However, one additional method can
be used on the TI-84 (MP) through the use of the tab keys.
Tab Key Entry: Press 1 q to access the Tab Key matrix menu. A screen such as Figure 4 will
appear. Use the arrow keys to select the rows and columns of your matrix. When both row and column
Commands and Operations. The TI-83+ family of calculators contains powerful built-in commands for
linear algebra. Before you use them, however, you need to understand how and why they work. These
commands and operations will be introduced in the projects as they are needed. The TI-83+ Guidebook
describes each of these commands.
You will use the +, –, *, and / keys to perform the algebraic operations of addition, subtraction,
multiplication and division, on numbers. Some of these operations can also be done on matrices, but
remember that the size of matrices play a major role in the definition of these operations. To add (or
subtract) two matrices of the same size, A and B, you enter A + B (or A – B) followed by the ¸ key.
To multiply two matrices, C and D, that are compatible for multiplication (the number of columns in C
located on the Web site. By the end of the course, you will be able to use your TI-83+ to solve interesting
problems that are completely impractical for hand calculations.
Transferring Data and Programs. You will need to transfer the data and programs to your TI-83+. You
can do this transfer in the computer lab or from another TI-83+ which has the data and programs already.
Running a Program: 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
Figure 6: Program menu Figure 7: 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
Plotting Statistical Data. You can plot statistical data on your TI-83+ and use the calculator to analyze
the data graphically and numerically. The following directions describe methods for inputting lists of
data into the calculator, and how to plot those data values. Your Guidebook will describe how to utilize
the numeric capabilities of the TI-83+.
Entering Lists Directly. Each list must be contained inside a pair of opening and closing “curly braces”.
Entering Lists in the STAT menu. Select S EDIT ¸ to enter a list. If a list is not empty, put
your cursor on a list name such as L1, and press C. Pressing the down arrow will delete entries in the
list.
Selecting the Plot. To plot statistical data you need to enter the data into two lists. Once the data has
Essentials for the TI-89 1
Essentials for the TI-89
Special Keys.
2 Press and release, then press the appropriate key to access
the 2nd (yellow) operation.
Entering Matrices: There are several ways to enter matrices into the TI-89. One method consists of
entering the matrix directly as an expression, another method is through the use of the matrix editor
found in the Data/Matrix Editor in the O menu. Below are examples that illustrate the creation of
Direct Entry: Begin a matrix with a square bracket, and each row of the matrix with a matching pair of
square brackets; separate the entries in the rows with commas. Figure 1 below shows how you can enter
the matrix
123
456
⎡
⎣
⎢⎤
⎦
⎥
directly and assign to it the name aa—the symbol ! appears after pressing the §
The Matrix Editor: Press O, scroll down to the Data/Matrix Editor and press ¸. See Figure 2.
You will be asked whether you want to work on the current data item, open a data item, or make a new
data item. Choose 3:New to construct a new matrix. See Figure 3.
Essentials for the TI-89 2
Use the right arrow to select either Data, Matrix, or List. Choose 2:Matrix by pressing ¸. Scroll
down to Variable, type in the name of the matrix, input number of rows and columns in the appropriate
places, and press ¸ twice. See Figure 4. If the variable name is not being used for anything else, the
TI-89 will allow you to start inputing entries directly into the matrix. See Figure 5.
Figure 4: Choose name, rows and columns Figure 5: Input the matrix
Commands and Operations. The TI–89 calculator contains powerful built-in commands for linear
algebra. Before you use them, however, you need to understand how and why they work. These
commands and operations will be introduced in the projects as they are needed. The TI-89 Guidebook
describes each of these commands.
Data Files. The 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.
Essentials for the TI-89 3
Running a Program: 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 in the program called ALINEAR. Press ¸ on the ALINEAR program, close the
parentheses and press ¸ again. The screen from Figure 6 should appear. Use the right arrow to
Plotting Statistical Data. You can plot statistical data on your TI-89 and use the calculator to analyze
the data graphically and numerically. The following directions describe methods for inputting lists of
data into the calculator, and how to plot those data values. Your Guidebook will describe how to utilize
the numeric capabilities of the TI-89.
Selecting the Plot. To plot statistical data you need to enter the data into two lists. Once the data has
been input, and you are still in the List editor, press „ for Plot Setup and ƒ to define the plot. Choose
the plot type, choose the mark, and set lists in x and y. See Figure 9.
TI Project: Systems of Linear Equations, Echelon Forms
The problems in this project will allow you to use row operation techniques and to reinforce the concepts
of row echelon form (REF) and reduced row echelon form (RREF). You will learn to make the transition
from equations to matrices and vice versa. Review Sections 1.1 – 1.5 of the text. Data for this project is
in the SYSTEMS section of the PROJDATA program.
Row Operations. The TI Calculator has three operations that correspond to the three elementary row
operations that appear in Section 1.1 of the text on page 7. (The row operation for adding two rows will
not be used in this project.)
TI-83+ and TI-84+ (MP)
Bring up the 2 > MATH menu and scroll down. To use the row operations efficiently it is convenient
to store matrices in memory.
rowswap(matrix,row1,row2)
Returns a matrix obtained from a given matrix by interchanging two rows. Enter the matrix, the
number of the first row to be interchanged, the number of the row with which it is to be
interchanged.
Time saving tip: For repeated row operations it is convenient to store the partial result into a named
matrix in memory. For example *row+(7,[A],1,3)![A] replaces A with a new matrix obtained from A
by replacing the third row with itself plus 7 times the first row. For another row operation of the same
type, press 2 [, your last entry *row+(7,[A],1,3)![A] appears on the screen; use the cursor keys
to move to the desired positions and edit entries, press ¸ and you will have performed another row
TI-89
Bring up the 2 I 4:Matrix J:Row ops menu and scroll down. To use the row operations
efficiently it is convenient to store matrices in memory.
rowSwap(matrix,row1,row2)
Returns a matrix obtained from a given matrix by interchanging two rows. Enter the matrix, the
number of the first row to be interchanged, the number of the row with which it is to be
interchanged.
Time saving tip: For repeated row operations it is convenient to store the partial result into a named
matrix in memory. For example mRowAdd(7,aa,1,3)!aa replaces A with a new matrix obtained from A
by replacing the third row with itself plus 7 times the first row. For another row operation of the same
type, use the cursor keys to move to the desired positions and edit entries, press ¸ and you will have
performed another row operation on A!
TI Project: Systems of Linear Equations, Echelon Forms 3
Problem 1. For each one of the following systems of linear equations:
x1 + 2x2 – x3 + 3x4 = –4 x1 + x2 – x3 = 7 x1 – 2x2 + x3 – x4 = 4
2x1 + 4x2 + 3x3 – x4 = 11 4x1 – x2 + 5x3 = 4 2x1 – 3x2 + 2x3 – 3x4 = –1
3x1 – 2x2 – 4x3 – x4 = –9 6x1 + x2 + 3x3 = 20 3x1 – 5x2 + 3x3 – 4x4 = 3
Problem 2. In a wind tunnel experiment, the force on a projectile due to air resistance was measured at different
velocities, according to the table below.
Velocity (100 ft/sec) 0 2 4 6 8 10
Force (100 lb) 0 2.9 14.8 39.6 74.3 119
Follow the indicated steps to find an interpolating polynomial for these data—that is a polynomial whose graph
passes through the data points given in the table. Use the function
p(t) = a0 + a
1t + a 2t2 + a
3t3 + a
4t4 + a
5t5.
(a) Use the statistical features of the calculator to plot the data in the table. Refer to the “Essentials for the
TI-xx” handout, which discusses plotting (statistical) data points.
(b) Explain how you would obtain a system of linear equations from the given data. Assume that t =
TI Project: Span and Linear Independence 1
TI Project: Span and Linear Independence
The problems will help you to reinforce the concepts of span and linear combinations. Review Sections
1.3-1.5 and 1.7 of the text. Data for this project is in the SPANLIN section of the PROJDATA program.
Entering Matrices. Refer to the “Matrix Essentials for the TI” handout, or consult the TI Guidebook.
Programs. You will need the GAUSS and BGAUSS programs to speed up your operations.
TI-83+ and TI-84+ (MP)
The output for the GAUSS and BGAUSS programs is in a matrix [J]. The programs were designed in this
form to facilitate the input if you execute the programs more than once. This will be the case if you are
attempting to use the programs to reduce a matrix to an echelon form. However, you must be careful
about the pivot positions in the output [J]—sometimes it will be necessary to swap rows of [J], and
rename the modified matrix as [J], then use [J] as the input. For example, to find an echelon form of the
matrix A =
−−
−
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1230
24 65
11 13
25 101
run GAUSS, then wait for the prompts:
MATRIX: prompts for a matrix, enter A.
ROW: prompts for a row, enter 1.
to replace [J] with the matrix obtained by swapping rows 2 and 4 of [J]. Run the program GAUSS and
use [J] as the input. You can quickly access matrix [J] by going to the matrix menu and pressing the up
arrow, (as opposed to pressing the down arrow repeatedly to get to [J]).
MATRIX: prompts for a matrix, enter [J].
ROW: prompts for a row, enter 2.
TI Project: Span and Linear Independence 2
−−
⎡
⎤
1230
TI-89
The output for the GAUSS and BGAUSS programs is in a matrix xx. The programs were designed in this
form to facilitate the input if you execute the programs more than once. This will be the case if you are
attempting to use the programs to reduce a matrix to an echelon form. However, you must be careful
about the pivot positions in the output xx—sometimes it will be necessary to swap rows of xx, and
The calculator displays the matrix xx =
−
−
⎣
⎢
⎢
⎢
⎦
⎥
⎥
⎥
0123
01 41
which is obtained from A by creating zeros in
the entries below the aa[1,1] entry. To continue to reduce A and obtain an echelon form, you cannot use
the matrix xx as the input, since there is a need for row swaps. Enter the command
rowSwap(xx,2,4)§xx
to replace xx with the matrix obtained by swapping rows 2 and 4 of xx. Run the program gauss and use
xx as the input.
Matrix: prompts for a matrix, enter xx
Row: prompts for a row, enter 2
TI Project: Span and Linear Independence 3
Problem 1. Let u =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
2
, v =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
1
and w =
1
2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
. Do as indicated below.
(a) Write down the definition of Span{u, v}.
(b) Suppose that you were asked the question: Is w in Span{u, v}? Use the definition of Span{u, v} to
write down what this means in terms of a vector equation.
(c) Write down what the question in (b) means in terms of a matrix equation.
(d) Suppose that you were asked the question: Are u, v, and w linearly independent? Use the definition of
Linear Independence to write down what this means in terms of a vector equation.
Problem 2. Let v1 =
1
4
2
0
⎡
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
, v2 =
1
1
0
3
−
⎡
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
, v3 =
−
−
⎡
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
1
6
2
6
, v4 =
2
1
4
9
−
⎡
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
. Establish that the set S = {v1, v2, v3, v4} is linearly dependent by