SYLLABUS
& LEARNING OUTCOMES
For the course syllabus, learning
outcomes and assessment details see the
course
web page and the
continuous assessment web page. At the end of this
module you'll be able to tackle questions such as those on the
2013
exam paper and those on the
2014
exam paper.
FORTNIGHTLY HOMEWORKS
The fortnightly homeworks count 40% towards the module assessment.
For the Science MA180 homeworks
click
here.
For the Maths & Education MA185 homeworks
click
here.
For the CS&IT MA190 homeworks
click
here.
To register
for homeworks use your eight digit ID number, and choose a memorable
password for the homework system.
Don't
forget
your
MA180/MA185/MA190 password because I am
unable to reset it
for you!
WEEKLY WORKSHOPS
Workshops begin on Monday 15th September. Details can be found
here.
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.
WHAT ARE THE EMPLOYMENT PROSPECTS
FOR A MATHS GRADUATE?
Have a look at the
links
here to answer this question.
ALGEBRA
MATERIAL
Algebra
text:
I can't find a single suitable book to recommend for the algebra
lectures. So
you'll just have to rely on the lecture notes and the various
references given below in the lecture outlines.
Algebra outline:
This module introduces the student to matrix algebra and systems
of equations, emphasizing that: (i) the entries of a matrix can be any
"numbers" for which we have a suitable notion of addition and
multiplication; (ii) matrix arithmetic underpins Ireland's
knowledge economy; (iii) matrix arithmetic over the "real numbers" has
a fruitful geometric interpretation. The module is divided into three
parts. Part I introduces a number system that will be new to many
students. Part II introduces matrix arithmetic over this number system,
as well as over the usual real number system. Part III develops a
geometric interpretation for matrix arithmetic and systems of equations
over the real numbers. Students will be expected to develop their
understanding of the topics through extensive calculation rather than
through formal theory.
Online Calculator:
This
online
calculator will help with all your modular arithmetic calculations.
Algebra lectures 201415:
The algebra lecture slides will be
uploaded to the web after each lecture and links to the slides will be
given below. A brief outline of each lecture will be added below
shortly after each lecture.
Here is some
student feedback after the first six weeks. I'll try to respond to
the criticisms as best I can.
1

Lecture
1:
Introduction to modular arithmetic. An application to the ISBN book
number was explained.
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

Lecture
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

Lecture
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

Lecture
4:
Deciphered an enciphered message sent from Agent 007.

5

Lecture
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

Lecture
6:
Introduced Euler's phi (or totient) function.
For more background on Euler's phi function take a look at the
wikipedia page
here.

7

Lecture
7:
Explained the RSA public key cryptosystem.
For more background on the RSA cryptosystem take a look at the
wikipedia page
here.

8

Lecture
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

Lecture
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.
Take a look at
this clip
for examples of matrix multiplication.

10

Lecture
10 (part 1) and
Lecture 10 (part 2):
Explained the notion of an affine matrix cryptosystem.

11

Lecture
11:
Deciphered a ciphertext obtained from an affine matrix cryptosystem. In
the process I got lots of practice of matrix multiplication.

12

Lecture
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

Lecture
13:
Explained 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.
I didn't get around to deriving the matrix representing rotation through an angle theta about the origin. See the slides of a previous year's lecture for this important derivation.

14

Lecture
14:
Illustrated the GaussJordan method for inverting a matrix. The method uses a sequence of row operations. 
15

Lecture
15:
Explained why the GaussJordan 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

Lecture
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. 
17

Lecture
17:
Proved that the determinant of a 2x2 matrix is equal to the area of a certain parallelogram up to sign. Then introduced and illustrated the notions of eigenvector and eigenvalue of a matrix. 
18

Lecture
18:
Explained how eigenvectors are involved in Google's page rank algorithm. (I intensionally over simplified the explanation. In particular, the importance I_{n} of a page is determined from the full network of pages on the internet and not just those [8 in my explanation] containg the given searched words.)
More details on the page rank algorithm can be found here.
Also stated and illustrated the important HamiltonCayley Theorem. 
19

Lecture
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 F_{n} = F_{n1} + F_{n2} for the number of rabbits in a field after n months, based on some assumptions about rabbit breeding. 
20

Lecture
20:
Talked about various occurences of the Golden Ratio. 
21

Lecture
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 F_{n} in the Fibonacci sequence. 
22

Lecture
22:
Used eigenvalues and eigenvectors to study a diseased population of frogs. 
23

Lecture
23:
Did some revision. 
24

Lecture
Sem II:
This is a Semester II lecture about the validity of logical arguments.

Algebra lectures 201213:
The course content is similar to that of 201213 and 201112. Links to
lecture slides from that year are available below.
Lecture
1
(201213): Introduction to modular aritmetic.
This lecture had to be repeated due to a timetable mixup. So these
slides are a compilation of the original lecture and the repeat
lecture.
Lecture
1 (201112): Introduction
to the course. Introduction to modular
aritmetic.
The notions of multiplicative inverse n^{1} and additive
inverse n modulo m were explained. The ISBN number was explained. The
lecture was short due to a power failure, so the IBAN was only
partially explained!
Take a look at the Wikipedia for a
good
introduction to modular arithmetic.
Lecture
2 (201213):
Lecture
2 (201112):
Explanation of a validation test for
International Bank Accounts
Numbers (IBANs) using arithmetic mod 97. Explanation of how the mod 97
value of a large integer can be calculated by hand. Introduction to
cryptography (Uboats, Bletchley Park, secure internet pages etc.) and
an example of a simple affine cryptosystem over a 26letter alphabet in
which message units are single letters. Some mod 26 calculations show
that the composite of the enciphering function f(n) with the
deciphering function f'(n) yields the identity function f'(f(n)) = n.
Lecture
3 (201213):
Attacked and decoded a
ciphertext
sent using an affine cryptosystem
with single letter message units. As part of the attack the Euclidean
algorithm was used to find the inverse of an integer n modulo m. A
careful analysis of this method for finding an inverse shows that a
number n has an inverse modulo m if and only if gcd(m,n)=1.
Lecture
3
(201112):
See here
for an obituary of Peter Hilton  Codebreaker and Mathematician. He
worked at Bletchley Park during World War II.
Lecture
4 (201213 & 201112):
A lady has a basket of eggs. If she
packs them in boxes containing
exactly 13 eggs she can pack all but 3 eggs. If she packs them in boxes
of 14 she can pack all but 6 eggs. If she packs them in boxes of 15 she
can pack all but 9 eggs. In order to determine the least possible
number of eggs she could have we used the Chinese Remainder Theorem.
(It's fun to use this theorem but a bit boring to state it. So we used
it without stating it!)
At the start of the lecture we computed the last digit of a large power
of an integer.
Take a look at the Wikepedia for a good
account of the Chinese Remainder Theorem.
Lecture
5 (201112): RSA public key cryptography and Euler's Phi function.
Lecture
6 (201112): RSA Cryptography and Euler's Theorem.
Lecture
5 (201213). Here are the photos of those who attended this
lecture on 27 September 2012.
Photo
1,
Photo
2,
Photo
3.
Lecture
6 (201213).
(
Some may also find it useful to look at the three
lectures 1,
2,
3
on RSA cryptography given in 201011.)
Lecture
7 (201112):
Deciphered a message sent to me using the RSA system. Explained how RSA
can be used to sign/authenticate documents electronically. Stated and
proved a special case of Euler's Theorem known as Fermat's Little
Theorem.
(Those who'd like to see a proof of the proposition (x^e)^d = x mod n
satisfied by the integers e,d,n used in RSA cryptography could take a
look at the second part of this
lecture which I gave last year.)
Lecture
8 (201112):
Recalled the notion of an mxn matrix (m rows, n columns). Recalled
addition/subtraction of two mxn matrices, and multiplication AB of an
mxn matrix A with a nxp matrix B. Recalled the notions of: zero matrix,
identity matrix, inverse matrix. From the definition of an inverse
matrix we evaluated the inverse of a 2x2 matrix modulo 5.
Lecture
9 (201112):
Gave the formula for the inverse of a 2x2 matrix, and indicated its
importance in affine encryption systems with message units of length 2.
Lecture
10 (201112):
Used the inverse of a 2x2 matrix modulo 37 to attack an affine
cryptosystem with 2letter message units. We finished off the attack
using a computer implementation of the cryptosystem.
Lecture
11 (201112):
Introduced the notion of a linear transformation and gave various
examples. Also gave an example of a transformations which is not
linear.
Lecture
12 (201112):
Proved that any linear transformation can be represaented by a matrix.
Stated (without proof) that any reflection in a line through the origin
is linear, as is any rotation about the origin. Noted that if S, T are
two linear transformations represented by matrices A, B, then the
composite transformation SoT: v >S(T(v)) is represented by the
matrix product AB. Derived the matrix representing an anticlockwise
rotation through an angle theta about the origin.
Lecture
13 (201112):
Illustrated the GaussJordan method for finding the inverse of a square
matrix. The method is based on three types of row operations: i) Add a
multiplie of one row to a different row; ii) interchange two rows;
(iii) multiply a row by a nonzero (i.e. invertible!!) scalar.
Lecture
14 (201112):
Explained how each of the three row operations can be represented as
premultiplication by a suitable matrix. Then proved that the
GaussJordan method always yields the inverse of a matrix when an
inverse exists. (The proof incorporates a hidden assumption which is
true but a bit tricky to prove. Please do try to spot the hidden
assumption!) Finished off by explaining how systems of
linear equations arise in real life, and how row operations can be used
to solve such systems.
Lecture
15 (201112):
Gave the definition and some basic properties of the determinant of a
2x2 matrix.
Lecture
16 (201112):
Proved that the determinant of a 2x2 matrix is equal to the area of a
parallelogram (up to sign). Then gave the definition of an eigenvalue
and eigenvector, together with examples and a method for computing
eigenvalues and vectors.
Lecture
17 (201112):
Motivated by a problem of mathematically modelling a field of breeding
rabbits, we used eigenvalues and eigenvectors to study the Fibonacci
sequence and the Golden Ratio.
Lecture
18 (201112):
Started with three equivalent definitions of the Golden Ratio, and
mentioned its role in aesthetics. The used eigenvectors to find a
closed formula for the nth term of the Fibonacci sequence.
Lecture
19 (201112):
Motivated by a problem of mathematically modelling an island of
infected frogs, the notion of a Markov matrix and Markov process was
introduced and used to study the long term prognosis for the island's
frog population.
Lecture
20 (201112):
Starting from a model of the internet as a collection of nodes, and
arrows from one node to another, we saw how the eigenvector
corresponding to the largest eigenvalue of a certain Markov matrix
gives the ranking produced by a Google search. The very important
Hamilton Cayley Theorem was squaezed in at the end of the lecture.
See this link
for a more detailed account of Google's page rank algorithm.
In Lectures 21 and 22 (201112) I worked through some old exam
questions (and
found a slip in one of them)!