## MA180-I & MA185 & MA190-I ALGEBRA (SEMESTER I) LECTURER: GRAHAM ELLIS

### 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 continuous assessment in Semester I consists of six fortnightly algebra/calculus homeworks of equal weight.

The continuous assessment counts 40% towards the module assessment for MA180-I and MA190-I. To pass MA180 or MA190 students must pass the year's continuous assessment consisting of 12 algebra/calculus homeworks.

For MA185 students the Semester I homeworks count 50% towards module MA187. The remaining 50% of module MA187 is based on six homeworks in Semester II.

Please click here to access the MA180, MA185 and MA190 homework sheets. (Due to a technical computer problem caused by a lightening strike this link will not work before Wednesday 16 September.) The first homework will be due on 2rd October. Late submissions will not be graded.

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 12th 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. You could also take a look at the lovely little book A Mathematicians Apology by G.H. Hardy which is available online here.

### WHAT ARE THE EMPLOYMENT PROSPECTS FOR A MATHS GRADUATE?

Have a look at the links here to answer this question.

### ALGEBRAMATERIAL

Algebra text:
Algebra & Geometry: An introduction to University Mathematics by Mark V.Lawson.
A pre-publication pdf version of this text is available on blackboard. This version is for private use only and the pdf version must not be made available on the internet.

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 2016-17:

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/modified below shortly after each lecture.

I'll place student feedback here.

### 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:
Began with the 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 Mathematicians Apology. This short book is well worth a read and is available online here.

Then 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.

Took attendance at todays lectures. Here are the photos: photo1, photo2, photo3.

### 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 :
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.

Took attendance at todays lectures. Here are the photos: photo1, photo2.

### 14

Lecture 14:
Illustrated the Gauss-Jordan method for inverting a matrix. The method uses a sequence of row operations.

### 15

Lecture 15:
Explained 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

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 In 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 Hamilton-Cayley 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 Fn = Fn-1 + Fn-2 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 Fn 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 24:
Did a bit more revision.