URL http://www.mast.queensu.ca/~helena/project.html

Math 112: Project


Warning - this page is still under construction

Introduction

20% of your final marks for this course will come from work on a project. This can be done individually, or in small groups. Each person will be required to write up a report on the work.

Although this is 20% of the marks for the course, you are not expected to write pages and pages. Quality counts more than quantity. About 2 pages of writing, in a clear style should be sufficient, together with additional work for mathematical formulas, calculations, graphs, diagrams, pictures.

Since this is a short course, we do not have time to go into all the details of applications of linear algebra. The purpose of the project is to give you an opportunity to look in more detail at some applications of the linear algebra we cover in the course.

I am hoping that some groups of students will give short presentation in class on their projects, so that we will all benefit from each others work. Giving a presentation is not compulsory, but if this is done, it will be taken into account in the marking of the project. Such a presentation could be about 10 minutes long.


Deadline

The project is due to be handed in on 25th June.

You are encouraged to have at least a rough draft ready by the last class, so that you are able to explain to others what your project is about, and if you hand in a draft at that time, I will mark it and give it back, so I can check that you are on the right track, and you will find out if there's anything else you need to add or improve.


Marking Scheme

The marks will be awarded based on the following points:

Note: the above marks add up to 25, but the project is marked out of 20. This means that if you do not do the presentation, it's still possible to get full marks. If your marks add up to more than 20, you'll be given a mark of 20 for the project.

I will also use the following checklist for marking:


Outline of suggested form of essay

Background description, overview:
Not so much detailed mathematical content needed here, mainly motivational or historical, etc. The start of the chapters of the text book gives a good example.
Mathematical ideas:
This part should explain the mathematical content, explaining in words the concepts, so that they can be easily understood.
Computations, formulas, or examples:
details of how these ideas are applied.
diagrams, graphs or pictures
Some visual component to the essay to convey the concepts in a clear way.

Note, you do not have to cover the above areas in the above order, eg, you could start with some diagrams, and then explain their meaning, importance, what they are about.


Method

Here is a suggested method of working:


Topics

The following is a list of possible topics to work on, with a brief outline of what would be expected. Other topics are also possible. Please let me know if you are thinking of working on something else.

Basically the idea is to take a section of the book that we will not be covering in class, and work through that section. Some possible topics are not in our text book; if you choose one of those, I will give you some appropriate reading material of a similar level to that of the text book.

Note on extra reading material

For a different view on some of the topics presented in the text book, you can look at sections from the following linear algebra text books:

the relevant sections are indicated below, and photocopies will be available on reserve in Straufer library, with the other extras mentioned below.

NOTE: you are not expected to read all of the extra material avaliable, it's just to give you more choice about what examples to use, and what applications to write about.

Markov chains:

Read: section 5.9

Write an essay about Markov chains. Describe what they are and how linear algebra is involved. Give several examples of their applications, and describe one example in detail.

Other material avaliable:

Powers of Matrices

Read: and

Write an essay on the applications of taking powers of matrices. Write about what kinds of things can happen, eg, does the matrix tend to infinity, or zero, or something else, when you keep multiplying it by itself? Describe the meaning of different kinds of behaviour in different problems. You can also describe the geometric interpretation. Calculate what happens for several examples, and make a table of some two by two matrices, and their limits under taking large powers. Also tabulate their determinant and trace. Are there any patterns? What is the relationship of powers of matrices to the eigen values?

Other material avaliable:

Linear Algebra in Economics

Read: section 1.3 and 3.7

Write an essay on applications of linear algebra to problems in economics. Include either a description of the Leontief method, or something else. Which ever case, write about possible applications, and describe one example in detail.

Other material avaliable:

Linear Algebra and Graph theory

Read: [BK]section 8.1, pages 417-434

Write an essay on the applications of linear algebra to graph theory and network problems. Write how to use these ideas to solve either the problem of the graph theory game (which I'll describe in lectures), or some other problem.

Other material avaliable:

Computer graphics

Read: section 3.8

Write an essay about the use of linear algebra in computer graphics. Include a description of homogeneous coordinates, what they are, and how they are used. Give examples of calculations and applications.

Other material avaliable:

Differential Equations

Read: see index of text book for various examples

If you have previously learned calculus, you could write an essay about solving differential equations using methods of linear algebra, eg, diagonalization. Include examples and write about applications.

Other material avaliable:

Recurrence relations

Read: section 5.8, also pages 89-90

We talked briefly about the Fibbonaci sequence in class. This is an example of a recurrence relation (also called a difference equation). You can write an essay explaining what a recurrence relation is, and giving some examples, eg, the Fibbonaci sequence. Use diagonalisation to find a solution for the nth term.

Other material avaliable:

Game theory

Read: [BK] section 8.11

Write an essay about how linear algebra Can be used in game theory. Explain what a pay off matrix is, and the concept of "saddle point". Give some examples of applications.

Other material avaliable:

Least squares

Read: section 7.5

Write an essay how the method of least squares is used to find best possible solutions to certain problems. give some examples of applications.

Other material avaliable:

Symmetry of two and three dimensional objects

Read: section 2.6

Describing the symmetries of an object is a very interesting question in mathematics and various branches of science. Matrices can be used to describe symmetries, eg, if we take a square, and put it's center at (0,0) in R2, then the matrcies which map the square to itself are those corresponding to rotations through 90 degrees, 180 degrees, and 270 degrees. We also have reflections in the lines y=0, x=0, y=x, and y=-x. So together with the identity matrix, there are 8 matrices that map the square to itself. We say it has a symmetry group of order 8. (order just means size). So this gives us a number that will tell us how symmetric the object is. We can look at the symmetries of other shapes, and see if they are more or less symmetric. We can also do this for three dimensional shapes, and for patterns that can be infinite.

For an essay on this topic, write about the concept of symmetry, how matrices measure this. Give some examples for various shapes, (preferably something more complicated than the square should be included, eg, cube).

Other material avaliable:

Note the book [CC] is a little bit advanced, so you may need to pick out which bits will be relevant, and ignore things you can't follow. I'll try and find something better if anyone would be interested.

Other ideas

More details

I'll add more detailed explicit guidelines here soon.


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If you have further questions, you can email me, or ask in class, or office hours.



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