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

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.

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.

- 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**. - Linear Programming I. Supplement.
- Linear Programming II. Supplement.
- Cubic Spline Interpolation. Handout.
- Leslie Matrices. Cryptography.
- Project Presentations.

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.

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).

- The Optimal Assignment Problem (The Hungarian Method).
**ELAAV**Chapter 11.4, Also, The Optimal Assignment Problem. D. Gale.**TFT**1981. 175--212. - 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. - Computer Graphics.
**ELAAV**Chapter 11.11, - Equilibrium (Temperature) Distributions.
**ELAAV**Chapter 11.12. - 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. 37--58. - 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. 217--246. - 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**.

- Continuous Time, Discrete State Space Markov Chains. F. Solomon.
- 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*. - 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, 406-410, 4.4
- 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, 63-72, 4.6
- 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, 4.5, 4.7
- 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, 9.10
- Graphs, Matrices, and Subspaces, Gilbert Strang, 24:1, 1993, 20-28, 3.1, 4.3
- 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, C, 4.3
- 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, 0.7
- 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

Copyright © 2005

Author: Kevin Mitchell (mitchell@hws.edu)

Last updated: JANUARY 2005

Department of Mathematics and Computer Science

Copyright © 2005

Author: Kevin Mitchell (mitchell@hws.edu)

Last updated: JANUARY 2005