Math 136 Project: Applications of Linear Algebra
The projects below are meant to expose you an application of linear
algebra. You can work alone or in groups of up to three people. Once you
or your group have chosen a topic, your first task will be to locate one
or two references that tell you how to use linear algebra to solve the
problem posed in the project. Sam and I will be available to help you
find references, etc.
It may be that you'd like to choose something that's not on this
list. If so, make an appointment to see me before Friday, May 9 so that
you can get started on a more detailed outline of your proposed
project.
Each group should send me an e-mail message specifying your topic and the
names of everyone in your group. The deadline for doing this is 5:00pm,
Friday, May 16. To avoid unnecessary duplication, please send me one
message per group (be sure to include the names of everone in the group).
Acknowledgement. I obtained the material below from Rebekah
Hahn, one of our former graduate students.
Project Descriptions
- Cryptography
-
Cryptography is the science of encoding and decoding messages and has
existed almost since man first started writing things down. More
recently, Cryptography is of interest to Computer Scientists and the
NSA. The idea behind this application is fairly basic in cryptography, but
already requires some good mathematics.
- Electrical Networks
-
Electrical Networks are used by Electrical Engineers to model
circuits. This application gives an introduction to modeling
circuits and how to use linear algebra to determine current or voltage
at a particular point in the circuit.
- Equilibrium Temperature Distributions
-
Equilibrium temperature distribution comes up in Physics and
Chemistry in trying to study how heat will eventually be distributed
across a thin metal plate, given the temperatures at the edge of the
plate. This application explores this idea, and introduces iterative
methods for finding solutions.
- Genetics
-
Genetics is the study of inheritance. In this application, you will
look at autosomal inheritance, the ideas behind recessive and
dominant traits, and how linear algebra can be used to determine how a
trait will be distributed in future generations.
- Linear Programming
-
Linear Programming is a technique used in Operations Research and has
applications to many different fields. In this application, you will
learn about the simplex method for maximizing or minimizing a certain
function, subject to a system of linear (in)equalities.
- Markov Chains
-
Markov Chains are used to model systems, like weather, which change
periodically. Typically, these changes are dependent on the immediate
history of the system, in the way that the weather one day depends on
conditions present the day before. This application nails down the
definition of Markov Chain and explores a few areas that they are used
in.
- Population Growth
-
Exponential functions aren't the only mathematical tools for studying
population growth. In this application, you will learn how linear
algebra can be applied to studying the growth of a female population
which has been divided according to age.
- Theory of Games
-
You may have heard of the infamous prisoner's dilemma. Game Theory is
the area of mathematics devoted to studying problems of this type,
which often come up in Economics. In this application, you'll find
out exactly what a game is and how to evaluate your chances of winning
based on the strategy you choose.
General Information
Each of you has to chose a topic for the project you will be doing this
quarter. As mentioned previously, you can work together in groups of
up to 3 people (though each person will have to turn in their own
report).
Due Dates
In general, these projects will take some time and effort on your
part. So, to keep you motivated and to help you get started early,
there will be two due dates for the project:
Wednesday, May 21, 2008: You need to turn in two references that
you have found and a summary (roughly a paragraph in length) of what
you've learned about your topic up to this point. This might include
definitions of words related to your topic or a discussion of how you
might start approaching one of the problems you've been assigned.
This is also the time you should let me know who you'll be working
with on your project. Not turning this in on time will affect your
final project grade.
5:00pm, Monday, June 2, 2008: Your report, in its final form, is
due. Not getting it in on time will affect your final project
grade.
Reports and Grading
Projects are worth 60 points, and each project should include the following
sections.
Background (15 points) This is mostly a discussion of how the
non-mathematical and mathematical portions of your topic fit together. In
other words, you need to talk about what you needed to know about your
topic in order to do the associated problems and how linear algebra fits
into the picture. So you might include the definitions of the words I've
given you, the linear algebra ideas you used (e.g. matrix multiplication,
solving linear systems, etc), and some explanation about why these ideas
were useful.
Solutions (35 points) You need to include solutions to the problems
included in this packet. Don't just give the answers, however.
Include a full, detailed explanation of what you're doing at each
step. You'll want to use words and write in full sentences, though
you can also have the occasional formula or sequence of equalities.
Bibliography (5 points) List the references you used to complete
this report. You don't need to get out your Strunk & White or anything,
just list title and author for any books you used. You should also include
a list of people that you consulted or any other form of help that you
received. For example, you might obtain some of your information from the
internet; in this case, you could include the website. You'll need at
least one book as a reference, preferably two, and a total of at least two
references.
You'll notice that there are still 5 points unaccounted for. The
remaining 5 points are for style: clarity, neatness, flow, design,
organization and creativity--it's important to be able to communicate
your ideas.
Note: you don't have to put your report in the precise order
given above. You may prefer to use the assigned problems to illustrate
how the ideas of the subject fit together with the mathematical ideas that
you will be using, in which case Background and Solutions would be
interwoven. Just make sure that these aspects appear in your report.