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.

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.

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).

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.

- 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.

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.

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.

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

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.

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

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:

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:

- [GW] section 2-6 pages 151-156

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.)

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:

- [BK] section 3.4, page 171-179
- [GW] section 4-2, page 237

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:

- [BK] section 8.6, pages 469-479
- [GW] section 3-7, pages 207-211 (This is about function spaces, not differential equations, but it may be of interest/use to refer to)

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)

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:

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:

- [BK] section 8.4, pages 450-461

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 R^{2}, 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:

- [CC] chapter 4, pages 109-117
- [CC] chapter 10, pages 292-293

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.

- 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
- The game described in lecture 6

- 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)

I'll add more detailed explicit guidelines here soon.

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

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