Math 214: Applied Linear Algebra

Offered:     Spring 2003
Instructor:  Kevin J. Mitchell

Office: Lansing 305 
Phone:  (315) 781-3619
Fax:    (315) 781-3860
E-mail: mitchell@hws.edu

Office Hrs:  M & W & F 8:30 to 9:30, M & W: 2:30 to 3:30, 
             Th 8:45 to 10:10. Often available at other times by appointment.

Class:       M-W-F 12:20 to 1:15 in GU 223

Readings:    Elementary Linear Algebra, 5th Ed. by Grossman
             Applications Supplement to Elementary Linear Algebra, by Grossman
             "Fire Control and Land Management in the Chaparral"
                  by Gearhart and Pierce
             "Markov Chains and Time Resources"
                  by Mitchell, Kolmes, and Ryan

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 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.

Course Goals

  1. to complete more of the theoretical development of linear algebra that was begun in Math 204,
  2. to develop a series of applications in class,
  3. learn to use Maple,
  4. to 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.

Course Materials

We will continue reading in Elementary Linear Algebra, 5th Ed. by Grossman, which you used for Math 204. This text is not available in the Bookstore; you should already have it. Additionally, there is a packet of applications for the course. Depending on how I am able to provide these materials, there may be up to a $10 charge to cover the copyright purchase and photocopying fees.

Using Maple

We will be using the computer algebra system Maple 7 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). Often I will be asking 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 New User's Tour in the Maple Help Menu or the Maple Essentials tutorial on N:. Be sure to copy it to your own desktop.

Assessment

Your course grade will be determined as follows: 
homework:                     20%     One or more graded assignments per week
2 one-hour exams (each):      20%     February 14 and March 21
final one-hour exam:          20%     Tuesday, May 5, 8:30AM
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. 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 Grossman's text in Chapters 1--3, and most of Chapter 5. 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.

  1. Introducing Maple. Review of the four fundamental spaces from Chapter 4.0
  2. More Maple: Review of scalar products and projections from Chapter 3 and Cofactors, Adjoints, and Inverses from Chapter 2.
  3. Change of Basis and Orthonormal Bases. Chapter 4.8, 4.9.
  4. Least squares: Fitting curves to data. Inner Product Spaces. Chapter 4.10, 4.11.
  5. Inner products, Angle, and Orthogonality. Complex Numbers. Chapter 4.11, Appendix 2.
  6. Eigenvalues and Eigenvectors. Population Models. Chapter 6.1, 6.2.
  7. Similar Matrices and Diagonalization. 6.3, 6.4.
  8. Regular and Absorbing Markov Chains. Supplement.
  9. Fire Control and Land Management in the Chaparral. W. B. Gearhart and J. G. Pierce. UMAP Module 687.
  10. Linear Programming I. Supplement.
  11. Linear Programming II. Supplement.
  12. Cubic Spline Interpolation. Handout.
  13. Leslie Matrices. Cryptography.
  14. Project Presentations.

Office Hours

My office is located in Lansing 305. My extension is 3619. My scheduled office hours are: 
M & W & F 8:30 to 9:30, M & W: 2:30 to 3:30, 
     Th 8:45 to 10:10. 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 mitchell@hws.edu.

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, 2003. My preferred date would be March 6 (the day before midsemester break or March 17 (the day you return from break). The projects are due no later than Tuesday, April 22, 2003. 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 PowerPoint presentation.

Key: ELAAV = Elementary Linear Algebra: Applications Version (7th Edition) by Anton and Rorres. ALA = Applications of Linear Algebra (3rd 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.

  1. The Optimal Assignment Problem (The Hungarian Method). ELAAV Chapter 11.4, ALA Chapter 6. Also, The Optimal Assignment Problem. D. Gale. TFT 1981. 175--212.
  2. Graph Theory. ELAAV Chapter 11.7, ALA Chapter 4. Also Determining the Reachability Matrix of a Digraph. R. Yarmish. TFT 1982. 497--515.
  3. Game Theory. ELAAV Chapter 11.8, ALA Chapter 5.
  4. Leontief Economic Models. ELAAV Chapter 11.9, ALA Chapter 8. Also General Equilibrium: A Leontief Economic Model. P. M. Tuchinsky. TFT 1980. 521--560.
  5. Management of Resources and Optimal Sustainable Yield. ELAAV Chapter 11.10. ALA Chapter 9.
  6. Computer Graphics. ELAAV Chapter 11.11, ALA Chapter 10.
  7. Equilibrium (Temperature) Distributions. ELAAV Chapter 11.12, ALA Chapter 11.
  8. Computed Tomography (CAT Scan). ELAAV Chapter 11.13, ALA Chapter 19. Iterative Reconstruction in Computerized Tomography. J. Harris and M. Kamel. TFT 1990. 151--176.
  9. Fractals. ELAAV Chapter 11.14.
  10. Chaos. ELAAV Chapters 11.14--15.
  11. Cryptography. ELAAV Chapter 11.16, ALA Chapter 18. Elementary Cryptology. J. F. Wampler. TFT 1993. 111=140.
  12. 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.
  13. Sustainable Harvesting. ELAAV Chapter 11.19 (and 11.18), ALA Chapter 14.
  14. Least Squares Approximations. ALA Chapter 15.
  15. 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")
  16. A Coordinate Transformation Problem in Air Traffic Control. H. Wang. The UMAP Journal Volume 2, Number 3: 1981. 9--16.
  17. A Model of Human Hearing. ELAAV Chapter 11.20 (and additional readings), ALA Chapter 16.
  18. 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.
  19. Spacecraft Attitude, Rotations and Quarternions. D. Pence. TFT 1984. 129--172.
  20. 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.
  21. The Sturmian Sequences for Isolating Zeros of Polynomials. A. M. Fink. TFT 1983. 37--58.
  22. 3-D Graphics in Calculus and Linear Algebra. Y. Nievergelt. TFT 1991. 125--170. Same article appears in TFT 1992. 125--170.
  23. Linear Programming Applications.
  24. Markov Chain Applications.
  25. From an 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.
  26. 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
Hobart and William Smith Colleges
Department of Mathematics and Computer Science
Copyright © 2003
Author: Kevin Mitchell (mitchell@hws.edu)
Last updated: JANUARY 2003