Offered: Spring 2003
Instructor: Kevin J. Mitchell
Office: Lansing 305
Phone: (315) 781-3619
Fax: (315) 781-3860
Office Hrs: M-W-F 10:30 to 11:30, M & W 2:15 to 3:00, Tues 1:30 to 3:00.
Often available at other times by appointment.
Class: M-W-F 9:05 to 10:00 in GU 206A
Readings: Elementary Linear Algebra, Applications Version, 8th Ed. by Anton and Rorres
"Fire Control and Land Management in the Chaparral"
by Gearhart and Pierce
"Markov Chains with Applications to Time Resources in Animals"
by Mitchell, Ryan, and Kolmes
About Math 214
Math 204 served as an introduction to the abstract concepts and theory of
vector spaces. The focus is on understanding the relations among vector spaces,
linear transformations, bases, dimension, linear systems, subspaces such as row, column
and null spaces, and determinants. On one level, Math 204 is a great introduction
to abstraction in mathematics because it is largely self-contained with few
mathematical prerequisites. For this reason, the subject plays a central role in
your development as a mathematics major.
However, another of the reasons that linear algebra plays such a central role
comes from our ability to interpret abstract concepts in various concrete
ways. Abstraction is what makes applied mathematics possible. For example, in Math 214
we will see
that vectors can be interpreted not only in the usual geometric way, but as
representatives of the possible "states" that a machine, an animal, or a process can have.
A matrix can be interpreted as the way that this machine, animal, or process
changes state. This is the theory of Markov chains. In a different situation,
vectors can be interpreted as codewords with matrix multiplication representing
the encoding and decoding processes. Matrix methods can be used to find lines or curves that
best fit a given set of data. The list of topics at the end of this handout
provides a host of additional applications, or better yet, "interpretations"
of linear algebra to real-world situations.
One difficulty with applied or real-world mathematics is that the data are often messy.
The component entries of a matrix or vector are not likely to be integers or simple fractions
as in most of the problems you have had to solve by hand in Math 204. Also, the dimensions of
the vectors and matrices or the size of the linear systems is likely to be larger than you have seen
previously. Imagine trying to row-reduce a complicated 10 x 10 matrix! Technology can help us
deal with more complicated data. We will use the Maple computer algebra software system to help
us manipulate such data.
The course will begin with an introduction to Maple which at the same time will serve to review
some of the core topics from Math 204. You will find that Maple is a tool
that you will be able to use in later mathematics
courses, whether or not the instructor actively incorporates into the class.
Complete more of the theoretical development of linear algebra that was begun in Math 204.
Develop a series of applications in class.
Learn to use Maple.
Investigate an application and complete a team-project with presentation.
The course culminates with your individual or two-person team project and
presentation of an application at the end of the term,
both orally to the class and in written
form. Using Maple should be a great help in this.
I have provided an extensive list of topics and resources for you at the end of this document.
But other topics are possible.
We will continue reading in Elementary Linear Algebra, 8th Ed. by Anton and Rorres,
which many of you used for Math 204.
Additional readings will be available to download from the
Math 214 folder on the N: drive.
You should download these materials when they become available and print them out for your use.
We will be using the computer algebra system Maple 9.5 that is available on the
campus network. I will be putting various files in a course folder entitled
Math 214 on the N: drive on the network. You should copy
these files to your own workspace on M:. There you can manipulate
your copies (and rename them). Sometimes I will ask you to turn in Maple
worksheets. This can be done by placing your files in the Dropbox
in the Math 214 folder. Make sure you are comfortable with this process. You may
want to try out the various New User's Tours in the Maple Help Menu if you have not
used Maple previously.
Your course grade will be determined as follows:
homework: 20% One or more graded assignments per week
2 one-hour exams (each): 20% February 18 and March 28
final one-hour exam: 20% Monday, May 9, 1:30AM
project and presentation: 20% Final week of class
Tests may well include a take-home or Maple component.
I also reserve the right to take class participation and attendance
into account in determining final grades.
Because of the nature of this course, its assignments, and its assessment, your
attendance and participation is crucial.
Mathematics is learned by regular, sustained, attentive
effort over an extended period.
Only when such effort has been invested will the concentrated study
for an exam have any benefit.
Therefore attendance at class are required.
Unexcused absences may adversely affect your grade;
certainly more than three absences will lower your grade. More
than six absences may result in your be dropped from the class.
If you must miss a class or lab for some reason beyond your control,
talk to me about it in advance.
Outline of Topics
I presume that you are familiar with the material in Anton's text in Chapters 1--5,
and most of Chapter 6. At least some of you have covered some of the material in Chapter 7 on
eigenvalues and eigenvectors. We will review some of the material in these two chapters, more
or less thoroughly as is necessary. The following gives a rough outline of the material in the
course, more or less in weekly units. It may prove useful to reorder some material. This
schedule is tentative at best, and is provided to give you a sense of what we will work
through. We may do more or less depending on your backgrounds.
Introducing Maple. Review of the four fundamental spaces from Chapter 5.0 (especially 5.5 and 5.6).
More Maple: Brief review of inner products and projections from Chapter 6.1 and 6.2.
Change of Basis and Orthonormal Bases. Chapter 6.3 and 6.5.
Least squares: Fitting curves to data; Fourier Series. Chapter 9.3 and 9.4.
Introduction to Eigenvalues and Eigenvectors. Complex Numbers. Chapter 7.1, 10.1--10.3.
Complex Vector Spaces. Chapter 10.4 and 10.5
Similar Matrices and Diagonalization. Chapter 7.2, 7.3, and 10.6
Regular and Absorbing Markov Chains. Supplement.
Fire Control and Land Management in the Chaparral. W. B. Gearhart and J. G. Pierce.
UMAP Module 687.
My office is located in Lansing 305.
My extension is 3619.
My scheduled office hours are:
Office Hrs: M-W-F 10:30 to 11:30, M & W 2:15 to 3:00, Tues 1:30 to 3:00.
I am often available at other times by appointment.
I am often in my office at other times of the day (e.g., before class),
and I encourage you to drop in to get hints or help with
course assignments or just to chat.
My e-mail address is
Project Information, Due Dates, Topics, and Resources
The final project for the course may be done with a partner or individually. No
two projects should cover the same topic. The project should be a substantial piece of
work (on the order of 10 or more pages). Since I do not expect that the projects
will be entirely original, you should be very clear in using appropriate referencing to all
sources of material that you use. A number of topics are suggested below.
A project proposal that includes a bibliography and rough outline is due on March 24, 2005.
My preferred date would be March 10 (the day before midsemester break or March 21 (the day you
return from break). The projects are due no later than the beginning of class on
Wednesday, April 27, 2005. The projects
should make use of Maple. In fact the entire project can be done as a Maple document and saved
in html format to make a web presentation possible. Other possibilities include using a
Key: ELAAV = Elementary Linear Algebra: Applications Version (8th Edition) by
Anton and Rorres. TFT = Tools for Teaching published by COMAP. Note the
specific years. The web version of this document contains links to a few of the sources.
Meyer = Matrix Analysis and Applied Linear Algebra by Carl Meyer (I have a copy).
The Optimal Assignment Problem (The Hungarian Method). ELAAV Chapter 11.4,
Also, The Optimal Assignment Problem. D. Gale. TFT 1981.
Graph Theory. ELAAV Chapter 11.7. Also Determining the
Reachability Matrix of a Digraph. R. Yarmish. TFT 1982. 497--515.
Graph Theory Applications to Electrical Networks. Meyer 200--205.
Game Theory. ELAAV Chapter 11.8. There are many other sources as well.
Leontief Economic Models. ELAAV Chapter 11.9. Also
General Equilibrium: A Leontief Economic Model.
P. M. Tuchinsky. TFT 1980. 521--560.
Management of Resources and Optimal Sustainable Yield. ELAAV Chapter 11.10.
Computed Tomography (CAT Scan). ELAAV Chapter 11.13.
Iterative Reconstruction in Computerized Tomography. J. Harris and
M. Kamel. TFT 1990. 151--176.
Fractals. ELAAV Chapter 11.14.
Chaos. ELAAV Chapters 11.14--15.
Age-Specific Population Growth (Leslie Matrix Model). ELAAV Chapter 11.18,
ALA Chapter 13.
Management of a Buffalo Herd. P.M. Tuchinsky. TFT 1981. 667--712.
Population Projection. Edward Keller. TFT 1980. 270--302.
Sustainable Harvesting. ELAAV Chapter 11.18 and 19.
What's Up Moonface? R. S. Strichartz.
The UMAP Journal Volume 6, Number 1: 1985. 9--16. (Determining the
tilt of the "Man in the Moon")
A Coordinate Transformation Problem in Air Traffic Control. H. Wang.
The UMAP Journal Volume 2, Number 3: 1981. 9--16.
A Model of Human Hearing. ELAAV Chapter 11.20 (and additional readings),
ALA Chapter 16.
Dimensional Analysis. Giordano, Wells, and Wilde. TFT 1987. 71--98.
The Use of Dimensional Analysis in Mathematical Modeling. Giordano, Jaye, and
Weir. TFT 1986. 171--194.
Spacecraft Attitude, Rotations and Quarternions. D. Pence. TFT 1984. 129--172.
Error Correcting Codes I. B. F. Rice and C. O. Wilde. TFT 1981. 501--526.
Aspects of Coding. S. Cohen. UMAP Unit 336. A Double-Error Correcting Code. G. J. Sherman.
UMAP Unit 337.
The Sturmian Sequences for Isolating Zeros of Polynomials. A. M. Fink. TFT 1983.
3-D Graphics in Calculus and Linear Algebra. Y. Nievergelt. TFT 1991.
125--170. Same article appears in TFT 1992. 125--170.
Linear Programming Applications.
Modeling Tomorrow's Energy System. T. O. Carroll. UMAP Expository Monograph Series. 1983.
A Linear Programming Model for Scheduling Prison Guards.
J.M. Maynard. TFT 1980. 389--428.
Markov Chain Applications.
Continuous Time, Discrete State Space Markov Chains. F. Solomon. TFT 1985.
Absorbing Markov Chains and the Number of Games in a World Series.
J. Brunner. The UMAP Journal Summer, 1987. 99--108.
Compartment Models in Biology.
R. Barnes. The UMAP Journal Volume 8, Number 2: Summer, 1987. 133--160.
The Cost Accounting Problem. D. O. Koehler. UMAP Unit 568.
online source Student
Projects in Linear Algebra (Instructor: David Arnold)The College Mathematics Journal is produced by the Mathematical
Association of America and contains articles accessible by college
students. You can find this journal in our library
in the periodical stacks. Each of the following is an article on linear
algebra selected from The College Mathematics Journal.
On Transformations and Matrices, Marc Swadener, 4:3, 1973, 44-51, 4.4
Binomial Matrices, Jay E. Strum, 8:5, 1977, 260-266
Mathematics in Archaeology, Gareth Williams, 13:1, 1982, 56-58, C
Visual Thinking about Rotations and Reflections, Tom Brieske, 15:5, 1984,
Harvesting a Grizzly Bear Population, Michael Caulfield and John Kent and
Daniel McCaffrey, 17:1, 1986, 34-46, 4.6, 9.10
Why Should We Pivot in Gaussian Elimination?, Edward Rozema, 19:1, 1988,
Rotations in Space and Orthogonal Matrices, David P. Kraines, 22:3, 1991,
245-247, C, 4.3, 4.4, 4.5
Number Theory and Linear Algebra: Exact Solutions of Integer Systems, George
Mackiw, 23:1, 1992, 52-58, 9.3
Gems of Exposition in Elementary Linear Algebra, David Carlson and Charles
R. Johnson and David Lay and A. Duane Porter, 23:4, 1992, 299-303, 1.2,
A Random Ladder Game: Permutations, Eigenvalues, and Convergence of Markov
Chains, Lester H. Lange and James W. Miller, 23:5, 1992, 373-385, 4.5,
Linear Algebra and Affine Planar Transformations, Gerald J. Porter, 24:1,
1993, 47-51, 0.4, 4.4
Iterative Methods in Introductory Linear Algebra, Donald R. LaTorre, 24:1,
1993, 79-88, 4.5, 9.6
How Does the NFL Rate the Passing Ability of Quarterbacks?, Roger W. Johnson,
24:5, 1993, 451-453, C
The Surveyor's Area Formula, Bart Braden, 17:4, 1986, 326-337, 5.2.6, 5.2.8
Cramer's Rule via Selective Annihilation, Dan Kalman, 18:2, 1987, 136-137,
Convex Coordinates, Probabilities, and the Superposition of States, J.N.Boyd
and P.N.Raychowdhury, 18:3, 1987, 186-194, 9.7
Determinantal Loci, Marvin Marcus, 23:1, 1992, 44-47, C
Roots of Cubics via Determinants, Robert Y. Suen, 25:2, 1994, 115-117,
Vectors Point Toward Pisa, Richard A. Dean, 2:2, 1971, 28-39, 6.3
Arithmetic Matrices and the Amazing Nine-Card Monte, Dean Clark and Dilip
K. Datta, 24:1, 1993, 52-56
A Geometric Interpretation of the Columns of the (Pseudo)Inverse of A,
Melvin J. Maron and Ghansham M. Manwani, 24:1, 1993, 73-75, C
The Matrix of a Rotation, Roger C. Alperin, 20:3, 1989, 230, C, 8.3
The Linear Transformation Associated with a Graph: Student Research Project,
Irl C. Bivens, 24:1, 1993, 76-78, 3.1, 9.1
Constructing a Map from a Table of Intercity Distances, Richard J. Pulskamp,
Systems of Linear Differential Equations by Laplace Transform, H. Guggenheimer,
23:3, 1992, 196-202, 6.2
Approaches to the Formula for the nth Fibonacci Number, Russell Jay Hendel,
25:2, 1994, 139-142, C, 0.2, 5.4.2, 9.3, 9.5
Connecting the Dots Parametrically: An Alternative to Cubic Splines, Wilbur
J. Hildebrand, 21:3, 1990, 208-215, 5.6.1, 9.6
Some Applications of Elementary Linear Algebra in Combinatorics, Richard
A. Brualdi and Jennifer J. Q. Massey, 24:1, 1993, 10-19, 3.2
There are many online sources which a search engine will reveal. In particular, you might
search for "linear algebra" along with "applications" and "Maple". For example,
there are a number of good applications
at David Lay's site.
Hobart and William Smith Colleges: Department of Mathematics and Computer Science