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 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.

Course Goals

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

Using Maple

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.

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 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.

  1. Introducing Maple. Review of the four fundamental spaces from Chapter 5.0 (especially 5.5 and 5.6).
  2. More Maple: Brief review of inner products and projections from Chapter 6.1 and 6.2.
  3. Change of Basis and Orthonormal Bases. Chapter 6.3 and 6.5.
  4. Least squares: Fitting curves to data; Fourier Series. Chapter 9.3 and 9.4.
  5. Introduction to Eigenvalues and Eigenvectors. Complex Numbers. Chapter 7.1, 10.1--10.3.
  6. Complex Vector Spaces. Chapter 10.4 and 10.5
  7. Similar Matrices and Diagonalization. Chapter 7.2, 7.3, and 10.6
  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: 
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 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, 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 PowerPoint presentation.

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).

  1. The Optimal Assignment Problem (The Hungarian Method). ELAAV Chapter 11.4, Also, The Optimal Assignment Problem. D. Gale. TFT 1981. 175--212.
  2. 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.
  3. Game Theory. ELAAV Chapter 11.8. There are many other sources as well.
  4. Leontief Economic Models. ELAAV Chapter 11.9. 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.
  6. Computer Graphics. ELAAV Chapter 11.11,
  7. Equilibrium (Temperature) Distributions. ELAAV Chapter 11.12.
  8. Computed Tomography (CAT Scan). ELAAV Chapter 11.13. 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. 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.
  12. Sustainable Harvesting. ELAAV Chapter 11.18 and 19.
  13. 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")
  14. A Coordinate Transformation Problem in Air Traffic Control. H. Wang. The UMAP Journal Volume 2, Number 3: 1981. 9--16.
  15. A Model of Human Hearing. ELAAV Chapter 11.20 (and additional readings), ALA Chapter 16.
  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.
  17. Spacecraft Attitude, Rotations and Quarternions. D. Pence. TFT 1984. 129--172.
  18. 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.
  19. The Sturmian Sequences for Isolating Zeros of Polynomials. A. M. Fink. TFT 1983. 37--58.
  20. 3-D Graphics in Calculus and Linear Algebra. Y. Nievergelt. TFT 1991. 125--170. Same article appears in TFT 1992. 125--170.
  21. Linear Programming Applications.
  22. Markov Chain Applications.
  23. 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.
  24. 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 © 2005
Author: Kevin Mitchell (mitchell@hws.edu)
Last updated: JANUARY 2005