Module Content

The module covers

  • Elementary number theory (8 lectures),
  • Matrix arithmetic (8 lectures),
  • Eigenvalues and vectors for 2x2 matrices (8 lectures),
  • and explains how these topics can be applied to

  • cryptography,
  • geometry/computer graphics,
  • Google page rank.

Module Coordinates

  • Lecturer: Prof Graham Ellis
  • Lectures: Wed 10am in AM200 and Thu 10am in AM200
  • Tutorials: Workshops begin on Monday 17th September. Details can be found here.
  • Recomended text: The lecture notes and web links (below) and continuous assessment problems contain all material necessary for the algebra section of this module. However, a supplementary algebra textbook available on Blackboard.
  • Problem sheet: available here.
  • Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard Algebra MA180/MA185/MA190(Semester I) pages. Blackboard will also be used for announcements and for posting grades.

Module Assessment

MA180 and MA190 students:

  • The end of semester exam is worth 60% of the total Semester I assessment. It will consist of three questions corresponding to the above three topics. A model exam paper is available here.
  • The continuous assessment is worth 40% of the total Semester I assessment. It will consist of six online problem sheets which will be made available here. Submission deadlines are strict. There are about 10 questions per problem sheet and to score 100% on the Semester I CA component you need to submit 50 or more correct answers.
A similar arrangement holds for MA180/MA190 in Semester 2 and in order to pass the module students must score a pass on the the year's continuous assessment and also score a pass on the weighted average of the year's continuous assessment and two end of semester exams.

MA185 students:

  • The end of semester exam is worth 100% of the total score returned for the MA185 module. It will consist of three questions corresponding to the above three topics. A model exam paper is available here.
  • The associated continuous assessment contributes towards 50% of the score returned for the MA187 Mathematical Skills module. The continuous assessment in Semester 2 contributes towards the remaining 50% of the MA187 assessment. The Semester I continuous assessment will consist of six online problem sheets which will be made available here. Submission deadlines are strict. There are about 10 questions per problem sheet and to score 100% on the Semester I CA component you need to submit 50 or more correct answers.

Supplementary Material and News

CLICKER OPINION POLLING may be used in some lectures.

WHAT IS MATHEMATICS?

I'm not too sure of the answer. But whatever it is it is possibly something a bit larger than what was taught in your school mathematics classes. If you are interested in the question then you should browse this article by Fields Medallist William Thurston. He won the Fields Medal for his work in geometry. You could also take a look at the lovely little book A Mathematician's Apology by G.H. Hardy which is available online here.

WHAT ARE THE EMPLOYMENT PROSPECTS FOR A MATHS GRADUATE?

Have a look at this link to answer this question.

PRACTICE HOMEWORK PROBLEMS

A few students have asked for more problems, similar to the homework problems, to practice on. I'll place some here after each homework has closed:

ADVANCED READING SUGGESTIONS

If you are finding the pace of lectures too slow and want to browse some advanced textbooks that cover advanced topics related to material in the lectures then you might take a look at

STUDENT FEEDBACK

I'll place student feedback here.

Lecture Notes

Lecture Notes
Lecture Summaries
1
Gave an informal introduction to modular arithmetic, and included an application to the ISBN book number.

For another introduction to modular arithmetic take a look at this Youtube clip. Then take a look at this clip, this clip and this clip
2
Explained Euclid's algorithm for finding the greatest common divisor of two numbers, and used it to find the inverse of some number n modulo m. An application of modular arithmetic to IBAN bank numbers was explained.

Take a look at this clip for another example of using the Euclidean algorithm to find the inverse of a number in modular arithmetic.

For more background on modular arithmetic take a look at the wikipedia page here.
3
Explained the basic ideas underlying cryptography. Discussed the Enigma machine and an affine cryptosystem on single letter message units.

For more background on the Enigma machine take a look at the wikipedia page here.
For more background on affine cryptosystems take a look at the wikipedia page here.
4
Deciphered an enciphered message sent from Agent 007.

Also explained about the homework system. The main points are:
- a deadline is a deadline.
- once you've chosen a password for your online homework, try your best to remeber it.
- there will be six homeworks this semester , each with 10 or more problems.
- If you get 50 or more problems correct you'll score 100% on the homework component.
- You can entrer an answer many times and the last entry is the only one that counts.

And one thing I forgot to say: if you use your friend's smartphone to input your answers, make sure that the phone is using your username and password and that it is not automatically remembering your friend's. This issue can cause the breakup of friendships!
5
Explained the Chinese Remainder Theorem.

For more background on the Chinese Remainder Theorem take a look at the wikipedia page here.
Also, take a look at this youtube explanation which uses easily calculated numbers,
6
Introduced Euler's phi (or totient) function. Also gave the definition of a public key cryptosystem

For more background on Euler's phi function take a look at the wikipedia page here.

I experimented with clickers and discovered that a majority of students would prefer to enter an answer such as "8" or "true/false" into a smart phone app rather then shout it out in real time. This experimentation meant that I didn't have enough time to finish the lecture. The last two slides in the uploaded lecture notes are what I had intended to write but didn't have time to write.
7
Explained the RSA public key cryptosystem.

For more background on the RSA cryptosystem take a look at the wikipedia page here.

It is worth considering, with the benefit of hindesight, the following quote

"both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate [number theory], and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean."

from G.H. Hardy's A Mathematician's Apology. This short book is well worth a read and is available online here.
8
Stated and illustrated Euler's Theorem. Then stated and proved a special case known as Fermat's little theorem.

For more background on Euler's Theorem take a look at the wikipedia page here.
For more background on Fermat's little heorem take a look at the wikipedia page here.
9
Introduced the notion of a matrix and the operations of addition, subtraction and multiplication.

For more background on matrix addition look at the wikipedia page here.

For more background on matrix multiplication look at the wikipedia page here.
10
Introduced the concepts of:
scalar multiplication for matrices,
an nxn identity matrix,
the inverse of an nxn matrix.

Derived a formula for the inverse of a 2x2 matrix.

Introduced the matrix affine cryptosystem.
11
Deciphered a ciphertext obtained from an affine matrix cryptosystem. In the process I got lots of practice of matrix multiplication.
12
Introduced the concept of a linear transformation of the plane. Showed that reflection in a line through the origin is a linear transormation.

For more background on linear transformations take a look at the Open Corseware notes from MIT here.
13
Explained why every linear transformation of the plane can be represented by a 2x2 matrix.

Stated that: (i) reflections in a line through the origin of the plane are linear transformation; (ii) rotations about the origin of the plane are linear transformations; (iii) the composite of any two linear transformations of the plane is linear.

Stated a theorem which asserts that compositiExplained why every linear transformation of the plane can be represented by a 2x2 matrix. Stated a theorem that asserts that composition of transformations corresponds to multiplication of matrices. Matrix multiplication has been invented just so that this theorem is true. on of linear transformations corresponds to multiplication of matrices. Matrix multiplication has been invented just so that this theorem is true.

Derived the matrix of a rotation of the plane about the origin.
14
Illustrated the Gauss-Jordan method for inverting a square matrix. The method uses a sequence of row operations.

See this Wikipedea page for more details on the method.

This lecture was delivered by Jerome Sheahan.
15
Rachel Quinlan delivered this lecture. Began with a review of row operations and the method for inverting a matrix. Her lecture slides are available here and here.

Finished up with a quick explanation of why the Gauss-Jordan method for finding the inverse of a matrix works.

Gave an example to illustrate that row operations can be used to solve systems of linear equations arising from "real life" problems.

For more background on systems of linear equations take a look at the wikipedia page here.
16
Defined the determinant and adjoint of a 2x2 matrix. Gave a formula for the inverse of a 2x2 matrix in terms of its determinant and adjoint. Explained that the determinant of a 2x2 matrix is equal to the area of a certain parallelogram up to sign.

This lecture was delivered by Jim Cruickshank.
17
Explained why det(AB) = det(A) det(B) for 2x2 matrices A, B. Then proved that the determinant of a 2x2 matrix is +/- the area of a certain parallelogram.

Introduced the definition of an eigenvector of a matrix A with associated eigenvalue. Gave some examples of eigenvectors and eigenvalues.

Noted that the matrix A of rotation through an angle (not equal to 0 or 180 degrees) has no eigenvectors.
18
Began with a rant about Big Brother and the dangers of Google, Facebook and other social media. The concerns are well documented on the internet: see for instance this page.

Then explained (in slightly oversimplified form) the idea behind Google's Page Rank algorithm which has led to its dominance as a search engine. In a nutshell, Google computes an eigenvector of a large matrix in order to decide the order in which it lists the results of a search.

Finished with the definition of the characteristic polynomial PA(x) of a 2x2 matrix A. Gave an example to illustrate the Cayley-Hamilton theorem which asserts PA(A)=0I.

Next lecture we'll use the characteristic polynomial to compute eigenvectors of a 2x2 matrix.

More details on the page rank algorithm can be found here.
19
Explained how to find eigenvalues of a 2x2 matrix using the characteristic equation. Explained how to find eigenvectors for the given eigenvalues.

Derived the recurrence relation Fn = Fn-1 + Fn-2 for the number of rabbits in a field after n months, based on some assumptions about rabbit breeding.
20
Rachel Quinlan talked about various aspects of the Golden Ratio. Her slides are here and here.
21
Explained how to express a suitable 2x2 matrix A in the form A=T-1 D T where D is diagonal. Here "suitable" means that A must have two eigenvectors such that the matrix T containing the two eigenvectors as columns is invertible.
Used the above expression to find a formula for the terms Fn in the Fibonacci sequence.
22
Used eigenvalues and eigenvectors to study a diseased population of frogs.
23
Went through some questions from last year's exam paper.
24