Linear algebra research topics

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

Math 112: Project

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Introduction

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.

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.

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Marking Scheme

The marks will be awarded based on the following points:

• 10 marks: Evidence that the material has been understood. Make sure your writing is clear and intelligible to any other member of class.
• 5 marks: Evidence that effort has been made to do some research or independent thinking, eg, reference to at least one other book or article than the text book (This could be something on the web), or making up your own examples (ie, not just copying everything from text book).
• 5 marks: Correct mathematics given in calculating examples.
• 5 marks: Presentation (to get full marks for presentation, you'll have to make sure this clearly explains all the main points of your project).

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:

• Make it clear what the essay is about.
• Clearly label and explain any graphs, diagrams, etc.
• Define all terminology and notation used.
• Explain briefly why mathematical results you use are true, or where they come from.
• Give references and clearly state results.
• Make sure the mathematics is correct.
• Use correct spelling, punctuation, grammar.
• In making calculations, make it clear
  • what the problem is,
  • How the math goes,
  • What the result is.

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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 picturesSome 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.

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Method

Here is a suggested method of working:

• Read the appropriate section or sections of the text book, and of the study guide.
• Read around a little if possible, or think up questions yourself, so you are not just copying word for word from the text book. (I can suggest things if you can't find anything, or are not sure what to use.)
• Make sure you understand what the topic is about, eg, by doing the exercises for that section of the book.
• Work out in detail and write up about some example that uses the ideas.

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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:

• [BK] Introductory Linear Algebra with Applications, by Bernard Kolman
• [CC] Linear algebra, and introductory approach by Charles Curtis
• [GW] Computaional linear algebra with models, by Gareth Williams

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:

• [BK] section 8-3, pages 439-449
• [GW] section 1-7, pages 60-73
• [GW] section

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

Write an essay on applications of linear algebra to problems in economics. Include either a description of the Leontief method, or something else. 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:

• [GW] section 1-8 pages 73-97
• "Scheduling conflict-free Parties for a dating service", Bryan L. Shader and Chanyoung Lee Shader, in the American mathematical Monthly, Feb 1997, Vol 104 #2. (This is probably a bit to much to look at properly, but it might give you an idea of some other applications.)

Computer graphics

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.

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.

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:

• [BK] section 8.7, pages 480-484 (This is on the Fibbonaci sequence)

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.

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.

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).

Markov chain topics: (5.9)

• In economics
• In sociology
• In biology

Leontief Economic models: (1.3, 3.7)

Game theory:

Computer graphics topics: (3.8)

Fractals

Graph theory topics:

• electrical networks
• applied psychology
• Topics in symmetry:
  • In chemistry, and crystals
  • In abstract geometry and relation to group theory.
• Linear programming:
• Reccurence relations: (2.7, 5.8)
  • Fibonacci sequence
• Applications to calculus:
  • Differential equations
  • Fourier analysis
• Mathematical investigations:
  • Investigation of powers of matrices.
  • Quadratic forms (8.2)