You will have two options for a project in this course. The first is to select a topic and study it. The second option is to solve a set of applied problems using the techniques you have studied in this course. You may collaborate on this project and I encourage you to work with other students, but each of you is responsible for submitting your own written report.

The project is intended to be an exploration in applications of linear algebra. The project should consist of a five to ten page paper (quality, not length is important) containing definitions, illustrative examples, what you have learned,  and about ten solved problems integrated with the narrative.

The presentation should be neat and clearly organized. Typing is preferred but not required. References and assistance must be credited. Feel free to additional references on any of these topics. You are also free to propose other topics.

 List of Possible Topics

1. Graphs and Networks A nice topic in mathematics which also has applications in physics, engineering, and computer science, among other fields. (Poole Section 2.4, Section 3.7, Strang 8.1)
2. Finite Linear Games. Games (or machines) where there are only a finite number of possible states (say Turing Machines from computer science or "Lights Out"). Analysis of the states can be done using matrices and modular arithmetic. (Poole Section 2.4)
3. Iterative Methods for Solving Linear Systems or for Computing Eigenvalues Different methods for solving linear systems--particularly useful when the matrix has lots of zeros. Includes the Jacobi method and the Gauss-Seidel method. (Poole Section 2.5, 4.5, Strang 9.3)
4. Markov Matrics and Ecomonic Models Very important in applied probability, although no knowledge of probability is required for this project. Markov models are important in physics, engineering, and biology in addition to economics. (Poole Section 3.7, 4.6, Strang 8.2)
5.  Linear Programming This is an approach to certain types of constrained optimization problems. It has particularly important applications in business and industry and is a central algorithm in operation research -- and are of applied mathematics dealing with planning and allocation problems.(Strang 8.3)
6.  Fourier Series
   Fourier series are especially useful in higher analysis and physics. It is valuable in statistics and electrical engineering.(Strang 8.4)
7. Computer Graphics
   The incredible computer graphics you see as special effects in movies and elsewhere is all made possible by linear algebra! (Strang 8.5)
8. The Fast Fourier Transform
   This algorithm and the simplex method in linear programming are two of the greatest achievements of applied mathematics in this century. This is why CD players, for example, are possible. This topic is extremely important in electrical engineering. (Strang 10.3)
9. Gaussian Elimination in Practice The actual application of Gaussian elimination on the computer involves details not covered in class. This gives you some insight into the problems people study in computational linear algebra and numerical analysis.(Strang 9.1)
10. The Singular Value Decomposition
    This is the very powerful and beautiful completion of the Fundamental Theorem of Linear Algebra. This decomposition is very important in computational linear algebra and statistics. (Strang 6.7)
11. Error-Correcting Codes. Using linear a;gebra to design binary codes that can not only detect transmission errors, but correct errors as well. (Poole Section 3.7)

References to start:

David Poole, Linear Algebra: A Modern Introduction, Brooks/Cole (2006).

Gilbert Strang,  Introduction to Linear Algebra, Wellesley-Cambridge (1998). (I have a copy in my office that may be borrowed for short periods of time.)

Article that might interest you (and should be available online from a campus computer at www.jstor.org)  are

1) Jozsef Denes and Gary L. Mullen, Rubik's Clock and Its Solution  Mathematics Magazine: Volume 68, Number 5, Pages: 378-381  1995

 In this note we describe Rubik's latest puzzle, Rubik's Clock, which is sold by Matchbox. Since the solution provides a nice application of linear algebra to a practical problem, we also provide a solution (which incidentally, is not provided at the time of purchase) to the puzzle. The "clock" consists of 18 small clocks, nine on each of two sides of a plastic device that also contains four buttons and four wheels, as illustrated in Figure 1 for the front face of the clock (the back is similar). Note that the four wheels are on the perimeter and the four buttons are centered about the central clock. 

2) Mathematics Magazine: Volume 53, Number 5, Pages: 269-276  1980

Linear Algebra in Geography: Eigenvectors of Networks

Philip D. Straffin, Jr.

 First Paragraph: Like economics and psychology before it, modern theoretical geography is a discipline in which the use of mathematics has become increasingly important. In this article I would like to discuss one use of linear algebra in geography. The application is elementary enough to be presented to a first undergraduate linear algebra class, although to my knowledge it has not appeared in linear texts except for a brief mention in [2]. It illustrates well the problem of giving meaningful interpretation to the results of mathematical manipulation of physical data. 

3) Mathematics Magazine: Volume 69, Number 1, Pages: 3-14  1996

Beyond Sin and Cos

Martin E. Muldoon and Abraham A. Ungar

First Paragraph: The higher-order circular and hyperbolic functions deserve to be better known. Here we give their main properties in order to make them more accessible to teachers and students in calculus, linear algebra and differential equations courses. The study of these functions can be related to such diverse topics as the binomial theorem and the fast Fourier transform. 

5) Mathematics Magazine: Volume 71, Number 4, Pages: 300-303  1998

Turning Lights Out with Linear Algebra

Marlow Anderson and Todd Feil

Lights Out is a puzzle played on a five-by-five array of lighted buttons. Each button is either on or off. When a buttom is pressed, it and its vertical and horizontal neighbors each change state: off to on, or on to off. One is given a configuration of buttons, some on, some off, and the goal is to turn all the lights off by pressing buttons. Using undergraduate linear algebra, we show that exactly one-fourth of the possible configurations are solvable, show how to recognize which configurations are solvable, and show how to solve those configurations that can be solved.

6) Mathematics Magazine: Volume 74, Number 1, Pages: 57-59  2001

An Easy Solution to Mini Lights Out

Summary: Mini Lights Out is a puzzle played on a four-by-four array of lighted buttons. Each button is either on or off. When a button is pressed, it and its vertical horizontal neighbors, including "wrapping" around the edges of the array, change state: off to on, or on to off. One is given a configuration of buttons some on, some off, and the goal is to turn all the light off by pressing the correct buttons. Using undergraduate linear algebra, we show that every possible configuration is solvable, and show an easy way to determine the solution. 

7) Mathematics Magazine: Volume 73, Number 2, Pages: 147--150  2000

Matrices, Continued Fractions, and Some Early History of Iteration Theory

Michael Sormani

 Summary: Undergraduate linear algebra is applied to a problem concerning the convergence of a given continued fraction. The original problem is first changed into a functional iteration problem and then into one involving matrix iterations. The matrix approach to the problem has an interesting history, connecting Charles Babbage, the designer of the first large mechanical computer, and Arthur Cayley, whose work on iterations is viewed as a primary factor in the development of the modern theory of systems. 

8) Mathematics Magazine: Volume 76, Number 2, Pages: 119-126  2003

A Natural Generalization of the Win-Loss Rating System

Charles Redmond

 Summary: How does one factor a team's schedule strength into its final standing? We offer a mathematically simple and natural way to do this that highlights some of the ideas covered in an elementary linear algebra course. Students learning about matrix multiplication, eigenvectors and eigenvalues, or even limits in a calculus course will find an interesting application to supplement their study.

10) College Math Journal: Volume 13, Number 1, Pages: 56-58  1982

Mathematics in Archaeology

Gareth Williams

 First Paragraph: The problem of placing sites and artifacts in proper chronological order is called sequence dating or seriation. An assumption made in archaeology is that graves which are close together in temporal order will be more likely to have similar contents than would graves which are farther apart in time. This note illustrates how archaeologists obtain information about the relative common contents and temporal ordering of graves. It can also be used to "dig up" interest in a linear algebra class. 

11) College Math Journal: Volume 24, Number 1, Pages: 20-28  1993

Graphs, Matrices, and Subspaces

Gilbert Strang

 First Paragraph: Graphs of the usual kind display functions. Graphs of a different kind (these have m edges connecting n nodes) lead to matrices. What y = f(x) is to calculus, matrices and subspaces are to linear algebra. Those are the central ideas, which students struggle with and to some extent master. This paper is about the incidence matrix A of a connected graph - and especially about its associated subspaces. 

12) College Math Journal: Volume 24, Number 1, Pages: 79-88  1993

Iterative Methods in Linear Algebra

Donald R. LaTorre

 First Paragraph: Introductory linear algebra is undergoing substantial change. Taught for several decades from the viewpoint of abstract vector spaces and linear transformations, it has now begun to respond to the widespread applicability of matrices in a variety of disciplines by moving towards a more matrix-oriented approach. This is a fundamental shift in perspective, brought about largely by the powerful advances in computational capability which have occurred in recent years. 

13) College Math Journal: Volume 26, Number 5, Pages: 358-360  1995

Linear Algebra on the Gridiron

Daniel C. Isaksen

 First Paragraph: Have you ever wondered how the intricate and precise marching patterns for a college football half-time show are created? Until just a few years ago, designing such shows was a time-consuming task requiring considerable technical skill. On a series of perhaps 30 pages, called "stuntsheets", the designer must record the band's formation on the football field at successive times in a show. Before computer software was developed to assist in making stuntsheets, labeled dots had to be placed by hand on a map of the field to show the locations of all band members at key moments. For a band of one hundred or more members this was a tedious process that required enormous care to make the transitions between successive stuntsheets uneventful. For example, one should avoid collisions -- maneuvers that assign two band members to the same spot at the same time! 

14) College Math Journal: Volume 32, Number 2, Pages: 128-134  2001

Image Reconstruction in Linear Algebra

Andrzej Kedzierawski and Olympia Nicodemi

 First Paragraph: Recently, we have been using one and two dimensional image reconstruction problems in our introductory course in linear algebra to motivate and illustrate various topics. We suppose that real world scenes are in "black and white" and that they come to us pre-digitalized as an array of pixels. Our hypothetical camera photographs a scene (an array) by matrix multiplication using a known matrix C. It transforms the original scene somewhat, perhaps blurring some of the details. Our main task is an inverse problem: to try to reconstruct the original scene from the photograph. The key to the classroom success of this application is a set of Maple programs we developed to produce visual images from digital data stored in arrays. These programs are available from the authors by e-mail. 

15) College Math Journal: Volume 32, Number 3, Pages: 208-210  2001

Linear Algebra in the Financial World

Barbara Swart

 First Paragraph: In the beginning of 1999 some financial funds bought call options on gold with an exercise price of $390 an ounce and exercise date December 1999. This means that they had the option to buy gold at $390 an ounce in December. The reason for entering into the contract was the expectation (based on the millennium fever and Y2K scare) that the price of gold would rise dramatically in December 1999. For example, if the price did rise to $400 an ounce, the pay-off would be $10 an ounce. The price of each call option was $1 per ounce, so the profit on 10,000 call options, would have been 10,000($400-$390) - 10,000($1) or $90,000! If the price remained below $390, the option would not be exercised and $10,000 lost. How was the price of $1 per call option determined so that the contract was fair to both parties? Can all possible contracts be priced fairly? These important problems involve concepts like arbitrage and complete markets, and this is what I want to discuss with the aid of linear algebra pictures as in [3]. 

Topic  selection due: November 19

Final Project  due: December 5. Noon.