Contact  Description  Topics  Texts  Notes  Lectures  Assessment and Assignments 
MAST20022 
Semester II 2020 
Lecturer: Arun Ram, 174 Peter Hall, email: aram@unimelb.edu.au
Time and Location: Zoom link on Canvas
https://canvas.lms.unimelb.edu.au/courses/87189
Lecture: Tuesday 13:0014:00,
Lecture: Thursday 9:0010:00,
Lecture: Friday 14:1515:15.
The lectures will be delivered live, online via Zoom. The Zoom link
is available through Canvas.
https://canvas.lms.unimelb.edu.au/courses/87189
If there are no technology glitches, each lecture will be recorded
and made available on Echo360 (accessible through Canvas) within 7 days
after the live lecture.
Watching the lecture videos is a very inefficient
use of time (and very incomplete)
if your goal is to do well on the exam.
A much better use of time, is to make handwritten copies, in your notebook,
of the writing in the video resources for this class that will be provided
via YouTube (see the YouTube links below).
You will do well on the exam if you are able to write solutions in the same
model as in the YouTube videos without the need for notes.
This is exactly the skill that will be assessed on the exam.
Zoom links for tutorials are available through Canvas.
https://canvas.lms.unimelb.edu.au/courses/87189/pages/listoftutorialsandzoomlinks?module_item_id=2174054
The tutorials will begin on Thursday of Week 1 (August 6).
In the first tutorial the "assignment marking exercise groups" will
be allocated. If you do not attend the first tutorial the tutor will assign
you to a group without your input.
Practice class: Monday 11:0012:00
Tutor: Weiying Guo
Practice class: Monday 15:1516:15
Tutor: Nemanja Poznanovic
Practice class: Monday 16:1517:15
Tutor: Kate Saunders
Practice class: Tuesday 14:1515:15
Tutor: Kate Saunders
Practice class: Thursday 12:0013:00 Tutor: Arun Ram
Practice class: Thursday 13:0014:00 Tutor: Arun Ram
Practice class: Friday 15:1516:15
Tutor: Nemanja Poznanovic
Practice class: Friday 16:1517:15
Tutor: Weiying Guo
A key part of Lecture time is the Ask me a question time. Generally this starts on the hour. With the challenges of Zoom, it might help if you feed questions to the student representatives, although this is not the only way (chat, unmuting and responding to my request for questions, and unmuting and just interrupting are also good).
Consultation hours: Arun Ram, Thursdays 25pm, Zoom link on Canvas,
email: aram@unimelb.edu.au
https://canvas.lms.unimelb.edu.au/courses/87189
Student Representatives: Gypsy Akhyar email: gakhyar@student.unimelb.edu.au and Dominique Cera dcera@student.unimelb.edu.au
Facebook group:
https://www.facebook.com/groups/378355086465467/
The lecturer and tutors will not be reading, looking at, or participating
in the Facebook group. If you wish to have the lecturer or tutors
read or participate, hold the discussion on Piazza
(which has math equation/LaTeX capability).
The Piazza link for this course
is available through Canvas.
https://canvas.lms.unimelb.edu.au/courses/87189
The homework assignments are at the following links: (total 7% each = 5% for the first submission +2% for the second submission)
The first deadline is for you to submit your solutions to the assignment questions. Your submission will be marked by the tutors. The first submission of each assignment is worth 5% of the total mark for the course.
After the first submission you will receive 4 assignments to 'mark' (the submissions of the others in you 'assignment marking group'). Only by reading solutions and trying to do marking yourself, do you learn how to optimise your marks on the exam. Read the solutions, provide feedback, and assign a mark for each question. You may be asked to provide justification for the mark you assign (so keep records of the marking scheme you used for each question). Submit your "marking" of the 4 assignments as the second submission of the assignment. Tutors will allocate marks for your second submission based on how conscientiously you undertake this exercise. The second submission of each assignement is worth 2% of the total mark for the course.
Because the assignment solutions will be released immediately after the due date/time, no late submissions will be accepted. If you miss a submission and you have a valid reason (medical certificate or equivalent), then you will be given the option to recover the marks for that assignment by making 3 distinct handwritten copies of the assignment solutions (the assignment solutions will be provided via YouTube videos) and one handwritten copy of the sample exam solutions.
It is recommended that you do and submit the assignments in the same way that you will for the exam: handwritten, a new page for each question, very clear exposition so that the marker can follow your argument easily, scanned with Camscanner, and uploaded question by question into Gradescope. The purpose of the assignments is to prepare your exam taking skills.
The handbook entry for this course is at https://handbook.unimelb.edu.au/2020/subjects/mast20022. The subject overview that one finds there:
This subject introduces the theory of groups, which is at the core of modern algebra, and which has applications in many parts of mathematics, chemistry, computer science and theoretical physics. It also develops the theory of linear algebra, building on material in earlier subjects and providing both a basis for later mathematics studies and an introduction to topics that have important applications in science and technology.
Topics include: modular arithmetic and RSA cryptography; abstract groups, homomorphisms, normal subgroups, quotient groups, group actions, symmetry groups, permutation groups and matrix groups; theory of general vector spaces, inner products, linear transformations, spectral theorem for normal matrices, Jordan normal form.
Assessment will be based on three written assignments due at regular intervals during semester amounting to a total of up to 50 pages (20%), and a 3hour written examination in the examination period (80%).
Almost everybody seems to agree that the "50 pages" doesn't make much sense in our modern technological world where some assignments might even be submitted in video form (the handbook entry has not been changed for more than 10 years) and a better estimate is to note that the handbook entry says that Total time commitment (should average) 170 hours. If we spread this over 14 weeks (12 class and 2 weeks exam study), we get 12.14 hours per week, and then if we subtract the number of class and tutorial hours per week (4) then we get 8.14 hours per week, and if we spread 12 weeks of 8 hours over 3 assignements then we get
4*8.14=32.6, as the average number of hours of study and work to complete each assignment.If you are spending more time than this, then please contact us immediately so that we can help you to make your study more efficient and effective so that you will be able to complete the necessary work for this class in this expected time commitment timeframe.
If you are spending less time than this, then maybe you should think twice about whether you will do well on the final exam.
The exam assesses whether you can write quality solutions to questions in an exam setting. So the main goal of the class is to learn to write quality, well presented solutions, that communicate well and thoroughly to the reader. Whether or not you get a correct answer has much less importance than whether your exposition is of good quality.
To help learn this skill, we will provide videos showing how to write quality solutions for:
The goal is to be able to write quality solutions in an exam setting.
A combination of factors, one of them being the Corona virus have made me start to think, early, how I am going to deliver my Group Theory and Linear algebra course next semester. Another major stimulus was the fact that Persi Diaconis is planning to teach a similar course starting at the end of March 2020, and I wanted to offer my notes to him as a resource.
The notes are structured in a form analogous to a dictionary. They are NOT meant to be read in a purely linear order starting from page one. The section on modules, does not require the section on groups, the section on Fields does not require the section on Rings. Unfortunately, physical parameters means that, for a published book one has to choose an ordering of the pages, and some people take this as a directive to read the pages in purely linear order. But this is an arftifice, and the reader that reads the pages in a linear order from page 1 will get a much lesser experience, and a significantly less valuable resource.
Happily, on a web page one can naturally interleave the links in a more stimulating fashion, making the whole exercise feel like a fun Sudoko puzzle:
Proof machine (How to do proofs)  
Example proofs  
Proofs: Fields  Examples: Groups  Groups 
Proofs: Vector spaces  Examples: Group actions  Group actions 
Examples: Rings  Fields  Proofs: Groups 
Examples: Modules  Vector spaces  Proofs: Group Actions 
Rings  Proofs: Rings  Examples: Fields 
Modules  Proofs: Modules  Examples: Vector spaces 
Group families

Commutative rings

Groups of low order

Other material can be put in different books. Sets and number systems will contain Osmosis topics and examples of rings and modules:
I had last taught the Group Theory and Linear Algebra course in 2011 and, going back to my web page for that course, I found an extensive and pleasant list of problems, and a complete set of lecture notes, handwritten and typed. The Problem lists were in html, and my old original notes needed conversion from Plain TeX to LaTeX. I have now done this and am now at a stage where I only needs to add pictures, clean up some proofs and proofread, proofread, proofread.
In my notes, the parallels between the structure theorems for Groups, Rings, Group actions, modules, Fields and Vector spaces are very carefully worked and presented. The examples for group families and groups of low order are well organised, thoughtful and thorough. I'm less happy with the fact that there is no good section on examples for rings, and no good section on examples for modules. But one step at a time, we shall see if we can put these in in due time. However, there is a very good argument that adding this material would make it too long to be a high impact resource for learning (noone ever really reads long books).
One standard addition to the course is to put in the Sylow theorems. For a long time I have felt that the Sylow theorems should be accompanied by the analogous theorems for Borel subgroups and tori (in algebraic groups) so that one sees vividly the analogies between $p$subgroups, Borel subgroups and tori. After some thinking about it, I decided that is is too much (both for the writer and reader) to put Borel subgroups and tori into this introductory text, and found that the Sylow theorems can very sensibly and motivationally included as exercises (with maps and directions for the proofs) to the seciton on Group Actions.
I struggle with where to place the material on sets, functions (morphisms in the cateogry of sets), cardinality (isomorphism classes in the category of sets), partially ordered sets, and ordered fields. When I teach this course, I often treat these topics in a couple of lectures entitled "Osmosis topics" (topics that you are somehow supposed to learn by osmosis. However, these will have to be properly treated in a different book, which be attractively titled Sets and Numbers or Numbers and Sets (which is better?).
If you don't communicate with us, then we can't help you. We'll do our best, but it is much easier if you take some responsibility too by asking questions and communicating in class and in tutorial.
The following problems page may have helpful examples: