TI Project: Cryptography
The problems in this project will help you to reinforce row operation techniques and to appreciate the
value of the inverse of a matrix and determinants in applications, such as message coding. Review
Sections 2.2, 3.1, 3.2 and Theorem 8 of Section 3.3 of the text. Data for this project is in the CRYPTO
section of the PROJDATA program.
TI-83+ and TI-84+ (MP)
Operations.
[A]*[B] Matrix multiplication (2 > Y* 2 > Ze). A and B are compatible for
multiplication.
det([A]) Determinant of the matrix A (2 >(right arrow to MATH menu) e 2 > Ye).
TI-89
Operations.
aa*bb Matrix multiplication (aa p bb ¸). A and B are compatible for multiplication.
mRowAdd(value,matrix,row1,row2)
Returns a matrix obtained form a given matrix by replacing a row with the sum of itself and a
multiple of another row. Enter the value to multiply the row by, the matrix, the number of the
row to be multiplied, the number of the row to be replaced.
Project:
You can make your own code by assigning each letter of the alphabet a positive integer from 1 to 26, and then
sending a message as integers instead of letter. For example, suppose that we assign numbers to the letters of the
alphabet according to the following table:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
4 10 14 22 2 17 6 3 18 11 15 5 25 19 26 23 8 1 21 12 9 20 13 24 7 16
You can e-mail a friend the message: HE SENT ME ROSES TODAY as
Since this code is not difficult to break, you want to use matrix multiplication to make it more secret. To put your
coded message into a matrix notice that there are 22 characters in it, including spaces between words; you can use
the number 0 for spaces, and ignore other punctuation marks if there are any in your messages. You can use the 3 × 8
numbers into it, so you added two “dummy” symbols at the end of your message. Let M =
−−
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
95 3
743
16 9 7
, so the
98 44 78 250 100 87 116 57
−−− −−
⎡
⎤
Problem 1. Let M be a matrix that is used in disguising a coded message.
(a) Describe a method for decoding the encoded message. You may use the example above for your
arguments. Use correct mathematical terms and notation.
(b) Suppose that M has integer entries and determinant ± 1. Explain why the entries of M-1 will also be
10 313747 30
31 1 7 4 2
101374 2 29
−− −
−−−−
−−−
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
Problem 2. Let N be the 4 × 4 identity matrix.
(a) Use your calculator to do a sequence of row replacement operations on N (using the command
described above for your calculator) so that none of the entries are 0, use only integers—positive and
negative—as your multipliers, store your final matrix in N. Write down N.
(b) Explain why it is true that det N = 1, then use your calculator to verify that this is indeed the case.
TI Project: Intersection of Two Vector Spaces
In this project you will have the opportunity to work with abstract concepts and review topics from linear
equations, basis and null space of a matrix. Problems of this type appear in the actuarial exams. Review
Sections 4.1 – 4.5 of the text. Data for this project is in the INTERVS section of the PROJDATA program.
TI-83+ and TI-84+ (MP)
Operations.
ref([A]) Computes a row echelon form of a matrix (2 > (right arrow to MATH menu) scroll down
to A:ref ¸ 2 >1 ¸).
rref([A]) Computes a reduced row echelon form of a matrix (2 > (right arrow to MATH menu)
scroll down to B:rref ¸ 2 >1¸).
TI-89
Operations.
ref(aa) Computes a row echelon form of a matrix (2 I 4:Matrix 3:ref( ¸).
Problem 1.
Let V = Span {v1,v2}, W = Span {w1,w2,w3}, where
v
1
=
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2
1
0
,
v
2
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
3
1
1
,
w
1
=−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
1
5
,
w
2
=
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
1
1
,
w
3
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
0
1
2
.
(d) Any subspace of Rn can also be represented as the set of solutions of a system of homogeneous linear
equations. For example, if U =
Span
1
0
3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
,
4
1
0
, then U can also be represented by an equation of the
1
⎡
⎤
4
⎡
⎤
⎣
⎢
⎢
⎦
⎥
⎥
1
Use the above technique applied to the vectors v1 and v2 to find a homogenous linear equation that
describes the vector space V. Write down the equation. Similarly find a homogenous linear equation
that describes W. Show all of your work and explain.
(e) Use the equations that you obtained in (d) to confirm that the answers that you gave in (c) are correct.
Problem 2. Repeat parts a-g of Problem 1, for V and W. V = Span {v1,v2, v3,v4}, W = Span {w1,w2,w3, w4}, where
v1 =
1
3
2
−
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
, v2 =
4
0
1
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
, v3 =
−
−
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
3
3
1
, and w1 =
4
3
0
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
, w2 =
2
3
3
−
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
, w3 =
1
2
2
−
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
, and w4 =
5
3
0
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
.
TI Project: Null Space and Column Space 1
TI Project: Null Space and Column Space
The problems in this project will help you to reinforce the concepts of null space and column space of a
matrix, and coordinate vectors. Review Sections 2.8 – 2.9 or 4.1 – 4.6 of the text. Data for this project is
in the NULLCOL section of the PROJDATA program.
TI-83+ and TI-84+ (MP)
Operations.
[A]*[B] Matrix multiplication (2 > Y* 2 > Ze). A and B are compatible for
multiplication.
TI-89
Operations.
aa*bb Matrix multiplication (aa p bb ¸). A and B are compatible for multiplication.
ref(aa) Computes a row echelon form of a matrix (2 I 4:Matrix 3:ref( ¸).
rref(aa) Computes a reduced row echelon form of a matrix (2 I 4:Matrix 4:rref(
¸).
TI Project: Null Space and Column Space 2
Problem. Let A =
243512
312018
00010
11026
13216
−
−
−− −−
−−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
.
(a) Use the
ref( command that is built into your calculator to find an echelon form of A. Write down this
echelon form in fractions, no decimals allowed.
(b) Write down the dimension of the column space of A and the dimension of the null space of A. Explain.
(c) Write down a basis for Col A. Explain.
(d) Write down a matrix (or matrices) that you would use to show that each one of the vectors
u =
3
2
1
4
3
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
, v =
11
4
1
2
3
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
, and w =
−
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
7
2
1
0
1
is in Col A.
(h) Use the
rref( command that is built into your calculator to compute the reduced row echelon form
of A. Write this matrix down (in fractions, no decimals allowed).
(i) Determine a basis for Nul A (in fractions, no decimals allowed). Explain.
(j) Use your calculator to show that each one of the vectors x =
5
1
2
0
1
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
, and y =
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
7
1
2
0
1
is in Nul A, and
determine their respective coordinate vectors, (in fractions, no decimals allowed) with respect to the
basis that you found in (i).
TI Project: Dynamical Systems 1
TI Project: Dynamical Systems
The problems in this project will show you how to apply the concepts of eigenvalue and eigenvector of a
matrix to an ecological dynamical system. Review Sections 5.1, 5.2 and 5.6 of the text. Data for this
project is in the DYNASYS section of the PROJDATA program.
TI-83+ and TI-84+ (MP)
Operations.
[A]*[B] Matrix multiplication (2 > Y* 2 > Ze). A and B are compatible for
multiplication.
augment Computes the augmented matrix [A B] (2 > (right arrow to MATH menu) 7) It
([A],[B]) requires two matrices, separated by a comma, a closing parenthesis, then press ¸.
TI-89
Operations.
aa*bb Matrix multiplication (aa p bb ¸). A and B are compatible for multiplication.
augment Computes the augmented matrix [A B] (2 I 4:Matrix 7:augment( ¸). It
(aa,bb) requires two matrices, separated by a comma, a closing parenthesis, then press ¸.
TI Project: Dynamical Systems 2
Problem 1. We want to model the situation in which three populations, owls, snakes and rats, compete against each
other. Let Oi, Si and R i denote, respectively, the number of owls, snakes and rats at time i in Woodland Park. Park
rangers have estimated that the populations of owls, snakes and rats can be described by the
equations:
Oi + 1 = .6 Oi + .2 Si + .4 R i
for i = 1, 2, 3, …
(a) If
xi =
O
S
R
i
i
i
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
is the population vector at time i. Write down a matrix A that describes this system.
(c) Use your calculator and the method you described in (b) to compute the population vectors xi, for the
periods i = 5, 10, 15, 20, 25, 30, 35, 40. Describe what you observe about the populations. Make a
graph to represent their growth (or decline); on the x-axis place the values of i, on the y-axis the
number of owls, snakes or rats.
(d) Use your calculator to compute the eigenvalues and eigenvectors of the matrix A. Write these down.
You can simplify your work by storing the matrix of eigenvectors as V.
(e) Use your calculator to represent the vector x0 as a linear combination of the eigenvectors of A. That is
Problem 2. The park rangers have been following the population of owls, snakes and rats in Woodland Park. At one
point they counted more snakes and rats than had been predicted for the well being of the three species in the park.
The rangers set traps to get rid of a few snakes and rats. After a careful study they noticed that the population of the
species could be described by the equations
Oi + 1 = .6 Oi + .2 Si + .4 Ri
TI Project: Dynamical Systems 3
(b) Let
x0 =
21
17
61
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
be the initial population vector with O0 = 21, S0 = 17, and R 0i = 61. Use your
calculator, and the method you used in Problem 1, to compute the population vectors xi, for the
periods i = 5, 15, 25, 35, 45, 55, 65, 75. Describe what you observe about the populations. Make a
graph to represent their growth (or decline); on the x-axis place the values of i, on the y-axis the
number of owls, snakes or rats.
TI Project: Gram-Schmidt Process and QR Factorization
The problems in this project will help you to reinforce the concepts of the Gram-Schmidt Process, and
the factorization of a matrix into an orthogonal matrix Q and an upper triangular matrix R. Review
Sections 6.1 – 6.4 of the text. Data for this project is in the GRSPROC section of the PROJDATA program.
Programs. You will need the PROJ and PROJV programs to speed up your operations.
PROJ Computes the orthogonal projection of the vector x onto the vector v. The program
prompts the user for the vectors as input.
TI-83+ and TI-84+ (MP)
Operations.
rref([A]) Computes a reduced row echelon form of a matrix (2 > (right arrow to MATH menu)
scroll down to B:rref ¸ 2 >1¸).
TI-89
Operations.
Problem 1. Let v1 =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
1
3
and v2 =
2
1
1
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
.
Problem 2. Let W =
{}
Span
x,x,x
123
, where
x
1
=−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
0
1
4
1
,
x
2
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
3
4
5
2
,
x
3
=−
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
0
1
2
5
. (The notation for this problem
follows the notation of the Gram-Schmidt process described in Theorem 11 of Section 6.4 of the text.)
(e) The Gram-Schmidt process produces formulas for computing v1, v2, and v3 in terms of
x1, x2, and x3. Explain how you can use these formulas to write the vectors x1, x2, and x3 in terms
of the vectors u1, u2, and u3. Write the vectors x1, x2, and x3 in terms of the vectors u1, u2, and
u3—DO NOT USE YOUR CALCULATOR.
(f) If A is the matrix that has the vectors x1, x2, and x3 as columns, use the information that you obtained
TI Project: Least-Squares Approximation 1
TI Project: Least-Squares Approximation
The problems in this project are applications of linear algebra to curve fitting and extrapolation. Review
Sections 6.5 and 6.6 of the text. Data for this project is in the LSAPPRX section of the PROJDATA program.
TI-83+ and TI-84+ (MP)
Operations.
[A]*[B] Matrix multiplication (2 > Y* 2 > Ze). A and B are compatible for
multiplication.
[A]-1 Computes the inverse of A (2 > Yie).
Graphing. Before attempting this project you should become familiar with the graphing capabilities of
TI-89
Operations.
aa*bb Matrix multiplication (aa p bb ¸). A and B are compatible for multiplication.
aa-1 Computes the inverse of A (aa Z · ¨ ¸).
Graphing. Before attempting this project you should become familiar with the graphing capabilities of
TI Project: Least-Squares Approximation 2
Problem 1. Wolverine World Wide, the manufacturer of Hush Puppies shoes, has experienced earnings per share
over a period of years as indicated in the following table.
Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
Earnings per share .02 .15 .31 .33 .28 .23 .42 .28 .49 .73 .94
a. Use your calculator to plot the data given above. (See the remarks on Graphing at the beginning of this
project.)
b. Set your viewing window as shown in Figure 1.
c. Find the equation
yB Bx
=+
01
of the least-squares line that best fits the data points from the previous
table. Follow the indicated steps.
(i) Enter the design matrix X, and the column vector y representing earnings per share into your
Problem 2. Find the equation
yB BxBx
=+ +
01 2
2
of the least-squares quadratic that best fits the data points from
the previous table. Follow the indicated steps.
(a) Enter the design matrix X, and the column vector y representing earnings per share into your
calculator.
(b) Use your calculator and follow the Examples in Section 6.6 of your text to compute the parameter
vector beta =
B
B
B
0
1
2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
. Write down the equation of the quadratic that best fits the data, use four decimal
places.
(c) Use your calculator to compute the predicted value yp = X*beta, the residual vector e = y – yp and its
norm n. Write these down (four decimal places), and compare with the information that you obtained in