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MAST20022
Group Theory and Linear Algebra

Semester II 2020

Lectures

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 13:00-14:00,
       Lecture: Thursday 9:00-10:00,
       Lecture: Friday 14:15-15:15.

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.

A susbstitute lecture for Group Theory and Linear Algebra in Semester II 2023 given by Arun Ram (main lecturer: Paul Zinn-Justin) Hand written Lecture Notes.

Tutorials

Zoom links for tutorials are available through Canvas.
https://canvas.lms.unimelb.edu.au/courses/87189/pages/list-of-tutorials-and-zoom-links?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 11:00-12:00 Tutor: Weiying Guo
       Practice class: Monday 15:15-16:15 Tutor: Nemanja Poznanovic
       Practice class: Monday 16:15-17:15 Tutor: Kate Saunders
       Practice class: Tuesday 14:15-15:15 Tutor: Kate Saunders
       Practice class: Thursday 12:00-13:00 Tutor: Arun Ram
       Practice class: Thursday 13:00-14:00 Tutor: Arun Ram
       Practice class: Friday 15:15-16:15 Tutor: Nemanja Poznanovic
       Practice class: Friday 16:15-17:15 Tutor: Weiying Guo

Discussion

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 2-5pm, 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

Assignments

The homework assignments are at the following links: (total 7% each = 5% for the first submission +2% for the second submission)

  • Assignment 1: Due 19 and 26 August before 4:00pm: Video solutions below
    • Question 1: gcd(864371,735577) https://youtu.be/MSK5Y3Sc3Vk
    • Question 2: Points on a circle in Z/4Z https://youtu.be/aiIVMsKcO7w
    • Question 3: Factorizing a polynomial in Z/8Z https://youtu.be/d9epxms6zyI
    • Question 4: The Euclidean algorithm
    • Question 5: Existence of the lcm -- Change + to intersection in this video existence and uniqueness of the gcd
      • Question 6: Characterisation of the lcm -- Change + to intersection in this characterisation of the gcd
    • Question 7: The field Q(i) -- See part of (c) this video Determining whether subsets of R and C are fields
    • Question 8: Decrypting an RSA message https://youtu.be/geoOwuXaYZE
  • Assignment 2: Due 16 and 23 September before 4:00pm
    • Question 1: Complex matrix Jordan form https://youtu.be/MYYtvIH3pIQ
    • Question 2: Matrix equation, diagonalizability and nilpotent https://youtu.be/tzGosF4vxSY
    • Question 3: Block decomposition over R https://youtu.be/RLbiCR7qYAY
    • Question 4: Jordan form possibilities for f5=f4 https://youtu.be/Fv96G5ulvZE
    • Question 5: Invertible elements in Z[sqrt{-5}] https://youtu.be/q1YGW6mrkSU
  • Assignment 3: Due 21 and 28 October before 4:00pm
    • Classifying Hermitian, unitary, normal and diagonalizable matrices https://youtu.be/VsbDkvtBPcA
    • The spectral theorem and applications https://youtu.be/EjYS7MMy284
    • The first Sylow theorem https://youtu.be/BSBuoM50hCk
    • The second Sylow theorem https://youtu.be/_ns55Oe7tZU
    • The third Sylow theorem https://youtu.be/zUORwBavCZg
    • The fourth Sylow theorem https://youtu.be/A8ANdqJKSXE
For each assignment submit your assigment solutions through Canvas/Gradescope before the first deadline and submit the marking exercise before the second deadline.

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.

Subject Overview

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.

Main Topics

  • (1) Greatest common divisors, Euclid’s algorithm, arithmetic modulo m.
  • (2) Definition and examples of fields, equations in fields.
  • (3) Vector spaces, bases and dimension, linear transformations.
  • (4) Matrices of linear transformations, direct sums, invariant subspaces, minimal polynomials
  • (5) Cayley-Hamilton theorem, Jordan normal form.
  • (6) Inner products, adjoints.
  • (7) Spectral theorem. Definition and examples of groups.
  • (8) Subgroups, cyclic groups, orders of groups & elements, products, isomorphisms.
  • (9) Lagrange's theorem, cosets, normal subgroups, quotient groups, homomorphisms.
  • (10) group actions, orbit-stabilizer relation, conjugation.
  • (11) Some results on classification of finite groups, Euclidean isometries

Assessment

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 3-hour 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 tutorial sheets
  • the assignments
  • one sample exam
  • various other sample exam problems that arise in lecture
We encourage students to make additional solutions and we will be happy to post them to make them available for all students.

Sample exams

  • Sample exam 1
    • A1. Show that a|b and a|c implies a^2|(b^2+3c^2) and Find gcd(323,377)
    • A3. Is Q(i) a field?
    • A4. The Jordan basis when the minimal polynomial is x^n
    • A5. Finding minimal polynomials and Finding Jordan form from minimal and characteristic polynomials
    • A6. Orders and subgroups in ℤ/10ℤ: This is done on pages 3 and 4 of Lecture Notes from 04.09.2020 (perhaps a couple of sentences more explanation written in words would be desirable.)
    • A7. Explaining why groups are nonisomorphic
    • A8. Quotients of ⟨(1,0)⟩ and ⟨(0,2)⟩ in Z/2Z x Z/4Z
    • A9. W⊆ (W⊥)⊥ and W=(W⊥)⊥ when dim(V) is finite
    • A10. Stabilizers and orbits for the GL(2) action on ℝ2
    • B1. Jordan normal form
    • B2. Size of SL2(Fp) and Sylow subgroups of SL2(Fp)
    • B3. Spectral theorem and B2=A.
    • B4. Actions and orbits of D_4 and Burnside's Lemma and an action of D4
  • Sample exam 2
  • Sample exam 3
  • Sample exam 4
  • Sample exam 5

The goal is to be able to write quality solutions in an exam setting.

Problem sheets

  • Problem sheet 1: Integers, modular arithmetic, gcd, Euclid's algorithm
  • Problem sheet 2: Fields, commutative rings, abelian groups, functions
  • Problem sheet 3: Vector space, bases, linear transformations
  • Problem sheet 4: Eigenvectors, minimal and characteristic polynomials
  • Problem sheet 5: Jordan normal form
  • Problem sheet 6: Inner products, adjoints, unitary and normal matrices
  • Problem sheet 7: Groups, homomorphisms and examples
  • Problem sheet 8: Normal subgroups and quotients
  • Problem sheet 9: Group actions, orbits, stabilizers and conjugacy
  • Problem sheet 10: 𝔼2 and isometries

Tutorial sheets

  • Tutorial sheet 1: Greatest common divisors, Euclid's algorithm, arithmetic modulo m
    • Common divisors of 56 and 72
    • Show that a|b and a|c implies a^2|(b^2+3c^2)
    • Find gcd(323,377)
    • Calculations in Z/10/Z, Z/7Z and Z/20Z
    • Powers of 3 mod 19
    • A condition for divisibility by 11
    • Show that if a=b (mod m) and b=c (mod m) then a=c (mod m)
    • Addition, multiplication and inverses in Z/7Z
    • The smallest positive number in 6Z+15Z
    • Show that if ac=bc (mod m) and gcd(c,m)=d then ad=bd (mod m)
    • Show that if p is prime then n^p=n mod p
  • Tutorial sheet 2: Fields, RSA cryptography
    • Determining whether subsets of R and C are fields
    • Show that, in a field, c0=0 and if ab=0 then a=0 or b=0
    • Algebraic closure, factoring and square roots in F_7
    • Inverses and solving equations in Z/35Z and Z/24Z
    • Using Fermat's little theorem for 3^{52} (mod 53)
    • Using Euler's theorem for 30^{62} (mod 77)
    • RSA encryption and decryption
  • Tutorial sheet 3: Bases, linear transformations, eigenvalues, direct sums, invariant subspaces
    • Is S={(1,3), (3,4), (2,3)} in (F5)2 a basis?
    • Are {1, sin2 x, cos2x} and {1, sin(2x), cos(2x)} linearly independent?
    • Show that {1, sqrt(2), sqrt(3)} is linearly independent over Q
    • Multiplication by a complex number as an R-linear transformation
    • (F5)4 as a direct sum of invariant subspaces , Second part
    • Eigenvalues and eigenvectors of a 3-cycle
  • Tutorial sheet 4: Minimal polynomials, diagonalisation
    • Using the minimal polynomial to find an inverse
    • Finding minimal polynomials
    • The Jordan basis when the minimal polynomial is x^n
    • The eigenvalues and eigenvectors of the transpose map and another part Perhaps these need to be patched together
    • Diagonalisability of f when f^4=1
    • Diagonalising f when f^2=f
    • Common eigenvectors when fg=gf
    • A linear transformation with no minimal polynomial
  • Tutorial sheet 5: Cayley-Hamilton theorem, Jordan normal form
    • Finding Jordan form from minimal and characteristic polynomials
    • Finding possible Jordan forms from the characteristic polynomial
    • Jordan forms, rank(A-alpha) and dimension of an eigenspace
    • Determining similar matrices
    • Characterizing Jordan form with subspaces
  • Tutorial sheet 6: Properties and examples of groups; subgroups, cyclic groups, orders of elements
    • Matrix groups and matrix non groups
    • Solving equations in groups
    • The group of nth roots of unity
    • Products of permutations in S6 WARNING: This video solution uses the opposite convention for permutation multiplication than was used in class. Depending on how it was presented (I have not reviewed the video recently) this video solution would probably still receive the bulk of the marks but, quite possibly, not full marks.
    • Orders of elements in S5 and Cx WARNING: This video has an incorrect solution to part (a)(vi), as the permutation (12)(345) in S5 (in cycle notation) has order 6. Depending on how it is presented (I have not reviewed the video recently) part (a)(vi) of this video solution would probably receive less than half of the marks allocated to part (a)(vi).
    • Cyclic subgroups of S3
    • If g2=1 then G is abelian
  • Tutorial sheet 7: Direct product, homomorphisms and isomorphisms, cosets
    • The order of (1,2) in Z/2Z x Z/8Z
    • Homomorphisms and nonhomomorphisms from GL(n) to GL(n)
    • Conjugation isomorphisms
    • Homomorphisms from Z to Z
    • Explaining why groups are nonisomorphic
    • Left cosets of ⟨ 3 ⟩ in Z/6Z
    • Left cosets of ⟨s,r2⟩ in the dihedral group D_6
  • Tutorial sheet 8: Normal subgroups, Lagrange's theorem, quotient groups
    • Quotients of ⟨(1,0)⟩ and ⟨(0,2)⟩ in Z/2Z x Z/4Z
    • Sizes of subgroups and intersections of subgroups
    • Quotient groups of cyclic groups are cyclic
    • If [G:H]=2 then H is normal.
    • The subgroups T, B and U in GL_2(R)
    • Determine all subgroups of the dihedral group D_5
  • Tutorial sheet 9: Inner products
    • Non inner products
    • Length of (1-2i, 2+3i) in ℂ2
    • Find an orthonormal basis of ℂ2 containing (1+i,1-i)
    • The complex parallelogram law for an inner product
    • The inner product ⟨ v,w ⟩ = Re(v,w)
    • The orthogonal complement of span{(0,1,0,1),(2,0,-3,-1)} in ℝ4
    • W⊆ (W⊥)⊥ and W=(W⊥)⊥ when dim(V) is finite
  • Tutorial sheet 10: Adjoints, spectral theorem
    • Self adjoint, isometric and normal matrices
    • Finding a unitary U such that U*AU is diagonal
    • Show that ker (f*) = (im f)⊥
    • Show that eigenvalues of self adjoint f are real and of isometries have absolute value 1
    • Show that every normal matrix has a square root
    • Show that there exists a unique decomposition A=B+C with B=B* and C=-C*
  • Tutorial sheet 11: Group actions, orbit-stabiliser theorem, Sylow theorems
    • Right multiplication by g-1 gives group action
    • Stabilizers and orbits of vertices and edge midpoints of a rectangle
    • Stabilizers and orbits for the GL(2) action on ℝ2
    • If |G|=9 and |X|=16 then an action of G on X has a fixed point
    • Conjugacy classes and centralisers of (12) and (123) in S3
    • 2, 3 and 7-Sylow subgroups in G when |G|=84
    • Show that there is a unique p-Sylow subgroup when G is abelian
    • Counting p-Sylow subgroups when |G|=30

Lectures and Tasks to keep up with the material 2020

The lectures will be recorded, but watching the recorded lectures is the LEAST efficient way to succeed in this course. The most efficient way to succeed in this course is to make handwritten copies, in your own handwriting of the solutions provided to sample exam questions -- provided in the links below.
  • Lecture 1, 4 August 2020: Invertible elements, multiples and the gcd   iPad chalkboard
    • The clock number system https://youtu.be/bG0URC2UR_A
    • Invertible elements https://youtu.be/vE4WubQg7-I
    • Multiples and the existence and uniqueness of the gcd
    Student TODO list:
    • Copy out example gcd question from pdf file
    • Copy out solution to this question from Tutorial 11 https://youtu.be/xbKEn0RgcH0
    • Copy out solution to this question from Sample Exam 1 https://youtu.be/72_3n9Q7py8
    • For something cool about multiplication in the clock number system
      watch this video https://www.youtube.com/watch?v=qhbuKbxJsk8
  • Lecture 2, 6 August 2020: The clock number system   iPad chalkboard
    • The relation between Z and Z/mZ: The Euclidean algorithm
    • Properties of Z/pZ when p is prime https://youtu.be/H_nGWeFFasc
    • The RSA cryptosystem https://youtu.be/vbGT8U1crfI
    Student TODO list:
    • Copy out the proof of the characterisation of the gcd
    • Read the notes on proof machine Proof machine (How to do proofs)
    • Do as much as you can of Tutorial sheet 1, to prepare for Tutorial: Tutorial sheet 1
  • Lecture 3, 7 August 2020:   iPad chalkboard
    • Field and algebraically closed field
    • Proof machine
    Student TODO list:
    • Do problem 1 on Assignment 1 and prepare it for upload to Gradescope (Unfortunately, Gradescope forces you to upload the whole assignment at once, so you cannot upload just a single question at a time. BE SURE TO PLAN SIGNIFICANT TIME TO FIGHT WITH GRADESCOPE before 4pm on 19th August. The Gradescope computer system will lock out all attempted uploads for Assignment 1 at 4pm on 19th August).
    • Copy out the definition of Ring and Field from the notes: Rings and Fields
    • Copy out the definitions of Group action, orbit and stabiliser, from the notes: Group actions
    Over the weekend (before lecture on Tuesday):
    • Do problem 4 on Assignment 1 and prepare it for upload it into Gradescope (Unfortunately, Gradescope forces you to upload the whole assignment at once, so you cannot upload just a single question at a time. BE SURE TO PLAN SIGNIFICANT TIME TO FIGHT WITH GRADESCOPE before 4pm on 19th August. The Gradescope computer system will lock out all attempted uploads for Assignment 1 at 4pm on 19th August).


  • Lecture 4, 11 August 2020: Osmosis topics   iPad chalkboard
    • The binomial theorem and the exponential function (see Sets and functions)
    • Ordered sets and ordered number systems (see The integers)
    • Sets, functions and equivalence relations (see Sets and functions)
    • Proof machine and Marksism (sections from Teaching Math in the Next Life)
    Student TODO list:
    • Do problem 5 on Assignment 1 and prepare it for upload to Gradescope.
    • Do problem 6 on Assignment 1 and prepare it for upload to Gradescope. (Unfortunately, Gradescope forces you to upload the whole assignment at once, so you cannot upload just a single question at a time. BE SURE TO PLAN SIGNIFICANT TIME TO FIGHT WITH GRADESCOPE before 4pm on 19th August. The Gradescope computer system will lock out all attempted uploads for Assignment 1 at 4pm on 19th August).
    • Copy out the solution to the question in this video: https://youtu.be/FVr2N2Ms5GI
  • Lecture 5, 13 August 2020: Vector spaces and bases   iPad chalkboard
    • Vector spaces and Subspaces (see page 11 of Linear Algebra)
    • Span, linear independence and bases (see page 12 of Linear Algebra)
    Student TODO list:
    • Do problem 7 on the assignment and prepare it for upload to Gradescope.
    • Copy out the solution to the question in this video: https://youtu.be/K7nbZlt-AoI
    • Make a handwritten copy of pages 4,5 of Lecture 5, 2011
    • Do as much as you can of Tutorial sheet 2 to prepare for this week's tutorial.
  • Lecture 6, 14 August 2020: Linear transformations   iPad chalkboard
    • Linear transformations (see pages 11, 12 and 13 of Linear Algebra)
    • The matrix of a linear transformation with respect to bases B and C (see page 13 of Linear Algebra)
    Student TODO list:
    • Do problem 2 on Assignment 1 and prepare it for upload to Gradescope.
    • Do problem 3 on Assignment 1 and prepare it for upload to Gradescope. (Unfortunately, Gradescope forces you to upload the whole assignment at once, so you cannot upload just a single question at a time. BE SURE TO PLAN SIGNIFICANT TIME TO FIGHT WITH GRADESCOPE before 4pm on 19th August. The Gradescope computer system will lock out all attempted uploads for Assignment 1 at 4pm on 19th August).
    • Over the weekend (before lecture on Tuesday):
    • Do problem 1 and problem 4 on Assignment 1 and prepare it for upload to Gradescope (Unfortunately, Gradescope forces you to upload the whole assignment at once, so you cannot upload just a single question at a time. BE SURE TO PLAN SIGNIFICANT TIME TO FIGHT WITH GRADESCOPE before 4pm on 19th August. The Gradescope computer system will lock out all attempted uploads for Assignment 1 at 4pm on 19th August).


  • Lecture 7, 18 August 2020: Eigenvectors and diagonalisability Handwritten lecture notes and ipad chalk
    • Eigenvectors and diagonalizability (see page 14 of Linear Algebra)
    • Eigenvectors and nullspaces https://youtu.be/ZkWg7zJMhw4
    • Distinct eigenvalues give linearly independent eignevectors https://youtu.be/TpgPzHNg1co
    Student TODO list:
    • Do problem 8 on Assignment 1. That should complete Assignment 1, which must be uploaded to Gradescope BEFORE 4pm on 19 August.
      (Unfortunately, Gradescope forces you to upload the whole assignment at once, so you cannot upload just a single question at a time. BE SURE TO PLAN SIGNIFICANT TIME TO FIGHT WITH GRADESCOPE before 4pm on 19th August. The Gradescope computer system will lock out all attempted uploads for Assignment 1 at 4pm on 19th August).
    • Do as much as you can of Tutorial sheet 3 to prepare for this week's tutorial.
  • Lecture 8, 20 August 2020: Handwritten lecture notes and ipad chalk
    • Eigenvectors and A-invariant subspaces https://youtu.be/3WE5FRFa2XQ
    • Determinant and characteristic polynomial of a diagonal matrix
    • Diagonalisability corresponds to a basis of eigenvectors
    • Minimal polynomials and The Cayley-Hamilton theorem
    Student TODO list:
    • Make a handwritten copy of the Proposition and the proof which appears on pages 4,5 6, 7 of pdf file)
    • Copy out the solution to this question from Sample exam 1 https://youtu.be/nKaRsjDQCCg
    • Calculate the determinant and the characteristic polynomial and the minimal polynomial of a diagonal matrix.
  • Lecture 9, 21 August 2020: Handwritten lecture notes and ipad chalk
    • Over algebraically closed fields an eigenvector exists https://youtu.be/ikQfn2M0qCM
    • Determinant, characteristic polynomial and minimal polynomial of a Jordan block
    Student TODO list
    • Copy out the solution to this question from Sample exam 1 https://youtu.be/F1l4Hp7yKvU
    • Make a handwritten copy of the proofs done in the YouTube videos for topics in Lectures 7, 8 and 9 (eigenvectors).
    Over the weekend (before lecture on Tuesday):
    • Do problem 5 on Assignment 2 and prepare it for upload to Gradescope.
    • Mark your 4 peer review assignments.


  • Lecture 10, 25 August 2020: Handwritten lecture notes and ipad chalk
    • Jordan normal form
    • Characteristic polynomial and minimal polynomial of a direct sum of Jordan blocks
    Student TODO list:
    • Finish the peer review exercise for Assignment 1 and send Canvas InBox messages with marked assignments as attachments (copy to Arun Ram, your tutor, and the student whose assignment it is).
    • Do Tutorial sheet 4 to prepare for this week's tutorial.
    • Make a handwritten copy of the solutions to Tutorial sheet 3 (video solutions are above).
  • Lecture 11, 27 August 2020: Handwritten lecture notes and ipad chalk
    • Semisimple and nilpotent matrices and their Jordan normal form
    • The characteristic polynomial and the minimal polynomial are conjugacy invariants
    • The subspaces ker((λ-A)i)
    Student TODO list:
    • Copy out the solution to this question from Sample exam 1 https://youtu.be/hivKEPYsnOU
    • Copy out the solution to this question from Sample exam 1 https://youtu.be/d-Ue9j-SMl4
    • Do question 1 on Assignment 2 and prepare it for upload to Gradescope.
  • Lecture 12, 28 August 2020: Handwritten lecture notes and ipad chalk
    • The evaluation homomorphism
    • Euclidean algorithm and gcd for polynomials
    • Relatively prime factors and block decomposition
    Student TODO list:
    • The proof of Proposition 1.6.12 on page 18 of Linear Algebra is very sketchy. Write out a proper, careful, proof that includes all the steps. (We also explained this proof in class, but it is a very useful thing to do to write the proof out carefully and think through how it relates to the tutorial questions on diagonalisability.)
    • Do question 5 on tutorial sheet 4 and then do question 2 on Assignment 2
    • Review the solutions for tutorial sheet 3 with a view towards making sure that if any of these problems appear on the final exam you'll be able to do them straightaway, in an exam setting.


  • Lecture 13, 1 September 2020: Handwritten lecture notes and ipad chalk
    • GLn (𝔽) and its conjugation action on Mn (𝔽)
    • Definitions of group, group action, subgroup
    • The examples GLn (𝔽), SLn (𝔽) and Sn
    Student TODO list:
    • Copy out the proof of the Chinese block decomposition (Proposition 1.7.3) from Jordan form Notes and organise it in proof machine so that all the steps are more clear.
    • Do Question 3 on Assignment 2 and prepare it for upload to Gradescope.
    • Do Tutorial sheet 5 to prepare for this week's tutorial.
  • Lecture 14, 3 September 2020: Handwritten lecture notes and ipad chalk
    • Order of a group, and order of an element
    • The subgroup generated by a subset S
    • Abelian group, and cyclic group
    Student TODO list:
    • Get a hot chocolate and sit down and peruse Groups of Low order for an hour and look at all the pretty pictures (it's better than reading the depressing news broadcasts!).
  • Lecture 15, 4 September 2020: Handwritten lecture notes and ipad chalk
    • Symmetric groups
    • Cyclic groups
    • Dihedral groups
    Student TODO list:
    • Do question 4 on Assignment 2 and prepare it for upload to Gradescope. That should complete Assignment 2!
    • Review the solutions for tutorial sheet 2 with a view towards making sure that if any of these problems appear on the final exam you'll be able to do them straightaway, in an exam setting.


  • Lecture 16, 8 September 2020: Handwritten lecture notes and ipad chalk
    • Direct products
    • Normal subgroups
    • Homomorphisms, kernels and images
    Student TODO list:
    • Make a handwritten copy of page 90 of Groups
      and practice doing the proofs of Propositions G.1.7 and G.1.8
      (see Propositions G.5.6 and G.5.7 of Proofs:Groups)
      enough times that you would be comfortable doing them if they appeared as questions on the exam.
    • Do Tutorial sheet 6 to prepare for this week's tutorial.
  • Lecture 17, 10 September 2020: Handwritten lecture notes and ipad chalk
    • Group actions, Stabilizers and orbits
    • Orbits partition the set
    • Relations between stabilizers
    Student TODO list:
    • Make a handwritten copy of page 93 of Group actions
      and practice doing the proofs of Propositions G.2.1, G.2.2 and G.2.3
      (see Propositions G.6.1, G.6.2 and G.6.3 of Proofs: Group Actions)
      enough times that you would be comfortable doing them if they appeared as questions on the exam.
    • Get a hot chocolate and sit down and peruse Groups of Low order for an hour.
      Given what has been covered in class now,
      see if you can make these examples make a bit more sense than when you perused them last week.
  • Lecture 18, 11 September 2020: Handwritten lecture notes and ipad chalk
    • Cosets
    • Cosets partition the group
    • Card(H)=Card(gH) i.e. diferent cosets of H are the same size
    • Card(G) = Card(H)⋅Card(G/H)
    Student TODO list:
    • Make a handwritten copy of Section G.1.1 Cosets on pages 88-89 of Groups
      and practice doing the proofs of Propositions G.1.2, G.1.3 and G.1.4
      (see Propositions G.5.1, G.5.2 and G.5.3 of Proofs:Groups)
      enough times that you would be comfortable doing them if they appeared as questions on the exam.
    • One of the necessary things for making a really good spaghetti sauce is a long simmer. We have covered most of the concepts needed for the proofs of the Sylow theorems and so it is a good time to prepare the scene for questions 3, 4, 5, 6 on Assignment 3 (don't do them yet, we just need to get the ingredients and start the simmer). Make a handwritten copy of each part of each of questions 3,4,5,6 on assignment 3, one part per page and notice what are the ingredients (see the example here Ingredients for simmer). After making this handwritten copy let these ingredients simmer in your brain for a week before you start to do these questions.


  • Lecture 19, 15 September 2020: Handwritten lecture notes and ipad chalk
    • Normal subgroups and quotient groups
    • G/N is a group if and only if N is a normal subgroup
    Student TODO list:
    • Make a handwritten copy of Section G.1.2 Cosets on pages 90 of Groups
      and practice doing the proofs of Theorems G.1.5 and G.1.6
      (see Propositions G.5.4 of Proofs:Groups)
      enough times that you would be comfortable doing them if they appeared as questions on the exam.
    • Do Tutorial sheet 7 to prepare for this week's tutorial.
    • Check on your simmer Ingredients for simmer and stir the sauce briefly by making another handwritten copy. You need to make sure your brain continues the (mostly unconscious) simmer on these ingredients.
  • Lecture 20, 17 September 2020: Handwritten lecture notes and ipad chalk
    • G/kerf ≃ imf.
    • The conjugation action: centralizers and conjugacy classes
    Student TODO list:
    • Make a handwritten copy of Section G.1.3 on pages 90 and 91 of Groups
      and practice doing the proofs of Proposition G.1.7, Proposition G.1.8 and especially Theorem G.1.9
      (see Theorem G.5.8 of Proofs:Groups)
      enough times that you would be comfortable doing them if they appeared as questions on the exam.
  • Lecture 21, 18 September 2020: Handwritten lecture notes and ipad chalk
    • Card(G) = Card(StabG(x))⋅Card(Gx)
    • The class equation
    Student TODO list:
    • Make a handwritten copy of Definition G.2.4, Proposition G.2.8, Proposition G.2.4 and Corollary G.2.5 of Group actions
      and practice doing the proofs of Propositions G.2.4, G.2.5
      (see Propositions G.6.4, G.6.5 of Proofs: Group Actions)
      enough times that you would be comfortable doing them if they appeared as questions on the exam.


  • Lecture 22, 22 September 2020: Handwritten lecture notes and ipad chalk
    • The conjugation action of G on G
    • The center Z(G)
    • The class equation
    Student TODO list:
    • Make a handwritten copy of section G.2.2, page 95, of Group actions
      and practice doing the proofs of Propositions G.2.8, G.2.9 and especially G.2.10
      (see Propositions G.6.8, G.6.9 and G.6.10 of Proofs: Group Actions)
      enough times that you would be comfortable doing them if they appeared as questions on the exam.
    • Do Tutorial sheet 8 to prepare for this week's tutorial.
    • We have now covered all material necessary for the proofs of the Sylow theorems. None of them is difficult, but the fourth is quite short to prove. Do Question 6 on Assignment 3 and prepare it for upload to Gradescope.
  • Lecture 23, 24 September 2020: Handwritten lecture notes and ipad chalk
    • Inner products
    • Orthogonals
    • Nonisotropic subspaces
    Student TODO list:
    • Make a handwritten copy of Sections 1.8.3 and 1.8.5 of Inner products
      and practice doing the proofs of Propositions 1.8.2 and 1.8.3
      (see the proofs of Propositions 1.8.10 and 1.8.11 on page 36) enough times that you would be comfortable doing them if they appeared as questions on the exam.
  • Lecture 24, 25 September 2020: Handwritten lecture notes and ipad chalk
    • Dual bases
    • Orthogonal projection
    Student TODO list:
    • Review the proof of Fermat's little theorem from https://youtu.be/H_nGWeFFasc AND the solution to question 11 on Tutorial sheet 1 at https://youtu.be/0JejVuWABqY
    • The proof of Fermat's little theorem uses binomial coefficients. Review these from Section 2.5 on page 37 and Section 2.7.5 on page 43 of Sets and functions notes.
    • Having reviewed Fermat's little theorem and binomial coefficients, do question 3 on Assignment 3 and prepare it for upload to Gradescope.
    • To put Sylow subgroups in context, identify the 5-Sylow subgroups and the 2-Sylow subgroups of D5 which appear in Question 6 on this week's tutorial sheet Tutorial sheet 8.


  • Lecture 25, 29 September 2020: Handwritten lecture notes and ipad chalk
    • Orthonormal bases
    • Gram-Schmidt
    Student TODO list:
    • Do Tutorial sheet 8 to prepare for this week's tutorial.
    • Do question 4 on Assignment 3 and prepare it for upload to Gradescope.
  • Lecture 26, 1 October 2020: Handwritten lecture notes and ipad chalk
    • Orthgonal block decomposition
    • Adjoints
    • Self adjoint, unitary, and normal matrices
    Student TODOlist:
    • Make a handwritten copy of Sections 1.8.6 and 1.8.7 of Inner products
      and practice doing the proofs of Propositions 1.8.4 and Theorem 1.8.4 and Theorem 1.8.6
      (see the proofs of pages 37 and 38) enough times that you would be comfortable doing them if they appeared as questions on the exam.
  • Lecture 27, 2 October 2020: Handwritten lecture notes and ipad chalk
    • Normal matrices produce both A and A* invariance
    • The Spectral Theorem
    Student TODO list:
    • Do Question 5 on Assignment 3 and prepare it for upload to Gradescope
    • It is STRONGLY recommended that over the break you also do questions 1 and 2 to finish off Assignment 3. Your other classes will have lots of work due in the last couple of weeks of semester and it would be good to have the assignment for this class done. We have covered all the necessary material for Assignment 3.
    • Make a handwritten copy of Sections 1.8.8 and 1.8.9 of Inner products
      and practice doing the proofs of Propositions 1.8.8 and Theorem 1.8.9
      (see the proofs of pages 39 and 40) enough times that you would be comfortable doing them if they appeared as questions on the exam.


  • Lecture 28, 13 October 2020: Handwritten lecture notes and ipad chalk
    • Proof of the Spectral theorem
    Student TODO list:
    • Do Tutorial sheet 10 to prepare for this week's tutorial.
    • Do questions 1 and 2 to finish off Assignment 3. Your other classes will have lots of work due in the last couple of weeks of semester and it would be good to have the assignment for this class done. We have covered all the necessary material for Assignment 3.
    • Make a handwritten copy of Sections 1.8.9 of Inner products
      and practice doing the proofs of Propositions 1.8.8 and Theorem 1.8.9
      (see the proofs of pages 39 and 40) enough times that you would be comfortable doing them if they appeared as questions on the exam.
  • Lecture 29, 15 October 2020: Handwritten lecture notes and ipad chalk
    • Conjugacy classes in the cyclic and dihedral groups
    • Burnsides Orbit theorem
  • Lecture 30, 16 October 2020: Handwritten lecture notes and ipad chalk
    • The conjugacy action on subgroups and normalizers
    • Review of the Sylow theorems
    Student TODO list:
    • It is time to start focused training for the exam: On Friday do questions B3 and B4 on Sample Exam 1. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
    • On Sunday do questions B1 and B2 on Sample Exam 1. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
    • On Sunday do questions A6, A7, A8, A9 , A10 on Sample Exam 1. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.


  • Lecture 31, 20 October 2020: Handwritten lecture notes and ipad chalk
    • The Quaternion group and Quaternions/Hamiltonians
    Student TODO list:
    • On Tuesday do questions A1, A2, A3, A4 and A5 on Sample Exam 1. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
    • On Wednesday do questions B2, B3 and B4 on Sample Exam 2. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
  • Lecture 32, 22 October 2020: Handwritten lecture notes and ipad chalk
    • Principal ideal domains and Euclidean algorthim for polynomials
    Student TODO list:
    • On Thursday do questions A7, A8, A9 and A10 and B1 on Sample Exam 2. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
  • Lecture 33, 23 October 2020:
    • GRAND FINAL EVE: No lecture
    Student TODO list:
    • On Friday do questions A1, A2, A3, A4 and A5 and A6 on Sample Exam 2. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
    • On Sunday do questions 11, 12, 13, 14, 15, 16 on Sample Exam 3. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
    • On Monday do questions 6, 7, 8, 9, 10 on Sample Exam 3. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.


  • Lecture 34, 27 October 2020: Handwritten lecture notes and ipad chalk
    • 𝔽[x] is a PID
    • Partial fractions
    Student TODO list:
    • On Monday do questions 1, 2, 3, 4, 5 on Sample Exam 3. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
  • Lecture 35, 29 October 2020: Handwritten lecture notes and ipad chalk
    • Matrices, traces and the trace inner product on matrices
    Student TODO list:
    • On Tuesday do questions B1, B2, B3, B4 on Sample Exam 4. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
    • On Wednesday do questions A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11 on Sample Exam 4. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
  • Lecture 36, 30 October 2020: Handwritten lecture notes and ipad chalk
    • Some remarks on row reduction
      • Student TODO list:
      • On Friday do questions B1, B2, B3, B4 B5 and B6 on Sample Exam 5. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
      • On Sunday do questions A1, A2, A3, A4, A5, A6, A7, A8, A9, A10 on Sample Exam 5. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
      • On each day of SWOTVAC do two tutorial sheets, or one tutorial sheet and one of the three assignments. Do each question 3 times, each time writing out the solution with small improvements to make it slightly better than the previous time you did it.
      • Be sure to sleep well and eat well, throughout, but particularly on the day before the exam.
      • Be sure you have your scanning and uploading to Gradescope working smoothly before the exam.


Lectures and lecture notes from 2011

  • Lecture 1, 26 July 2011: The clock and invertible elememnts
    Math Grammar: Definitions, Theorems and How to do Proofs (pdf file) and handwritten lecture notes-pdf file

    Examples of proofs written in proof machine (pdf file)

  • Lecture 2, 27 July 2011: gcd and Euclid's algorithm - handwritten lecture notes - pdf file
  • Lecture 3, 29 July 2011: Equivalence relations - handwritten lecture notes - pdf file

  • Lecture 4, 2 August 2011: Functions - hand written lecture notes - pdf file
  • Lecture 5, 3 August 2011: Rings and Fields (pdf file)
  • Lecture 6, 5 August 2011: C[t], gcd and Euclid's algorithm (pdf file)

  • Lecture 7, 9 August 2011: Vector spaces and linear transformations (pdf file)
  • Lecture 8, 10 August 2011: Span and bases (pdf file)
  • Lecture 9, 12 August 2011: Change of basis (pdf file)

  • Lecture 10, 16 August 2011: Eigenvectors and annihilators (pdf file)
  • Lecture 11, 17 August 2011: Minimal and characteristic polynomials (pdf file).
  • Lecture 12, 19 August 2011: Jordan normal form (pdf file)

  • Lecture 13, 23 August 2011: Block decomposition (pdf file)
  • Lecture 14, 24 August 2011: Cayley-Hamilton theorem (pdf file)
  • Lecture 15, 26 August 2011: Inner products and Gram-Schmidt (pdf file)

  • Lecture 16, 30 August 2011: Orthogonal complements and adjoints (pdf file)
  • Lecture 17, 31 August 2011: The spectral theorem (pdf file)
  • Lecture 18, 2 September 2011: Groups and group homomorphisms (pdf file)

  • Lecture 19, 6 September 2011: The polar decomposition (pdf file)
  • Lecture 20, 7 September 2011: Symmetric groups and subgroups generated by a subset (pdf file)
  • Lecture 21, 9 September 2011: Cyclic groups and products (pdf file)

  • Lecture 22, 13 September 2011: Cosets and quotient groups (pdf file)
  • Lecture 23, 14 September 2011: Quotient groups (pdf file)
  • Lecture 24, 16 September 2011: G/kerf≃imf (pdf file)

  • Lecture 25, 4 October 2011: Group actions, orbits, stabilizers (pdf file)
  • Lecture 26, 5 October 2011: Centres and p-groups (pdf file)
  • Lecture 27, 7 October 2011: Proof of the Orbit-Stabilizer theorem (pdf file)

  • Lecture 28, 11 October 2011: The affine orthogonal group and isometries (pdf file)
  • Lecture 29, 12 October 2011: Isometries of E2 (pdf file)
  • Lecture 30, 14 October 2011: Matching the affine orthogonal group with isometries (pdf file)

  • Lecture 31, 18 October 2011: Revision: Analogies (pdf file)
  • Lecture 32, 19 October 2011: Revision: The Fundamental Theorem of Algebra (pdf file)
  • Lecture 33, 21 October 2011: Revision: Proof machine (pdf file)

  • Lecture 34, 25 October 2011: Revision: Working sample randomly chosen problems
  • Lecture 35, 26 October 2011: Revision: Maths, Music and the Weil conjectures
  • Lecture 36, 28 October 2011: Revision: C[t]-modules (pdf file)

The little book around Structure and Action

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
  • Cyclic groups
  • Dihedral groups
  • Symmetric groups
  • Alternating groups



 Commutative rings
  • Fields, integral domains,
    fields of fractions
  • Euclidean domains, PIDs and UFDs
  • Polynomial rings



 Groups of low order
  • ℤ/2ℤ, the cyclic group of order 2
  • ℤ/2ℤ ×ℤ/2ℤ, the Klein 4 group
  • S3 ≅D3 , the nonabelian group of order 6
  • The dihedral group D4 of order 8
  • The tetrahedral group A4
  • The octahedral group S4
Already, because this includes proofs written in proof machine, this is totalling to about 160 pages. If we add the exercises for these topics we have somewhere in the neighborhood of 200 pages. That is probably enough for a single book.

Other material can be put in different books. Sets and number systems will contain Osmosis topics and examples of rings and modules:

  • Sets: subsets, unions, intersections, products
  • functions: injective, surjective, bijective, images
  • equivalence relations, partitions, fibers of functions
  • partially ordered sets, Zorn's lemma, supremum, infimum
  • ordered fields, inequalities
  • Cardinality

  • The positive integers, the nonnegative integers, the integers, and ideal in the integers
  • The fields ℚ, ℝ and ℂ.
  • The p-adic numbers ℤp and ℚp
  • The polynomial ring ℂ[ε], i.e. the group algebra of ℤ ≥0, and a study of its ideals
  • The ring ℂ[ ε, ε-1] i.e. the group algebra of ℤ, and some study of its ideals
  • Matrices, direct sums of matrix algebras, and ideals in direct sums of matrix algebras
Then a Linear algebra book can focus on orbit problems for subcategories of vector spaces:
  • ℂ-modules (vector spaces, bases, linear transformations, functors: tensor powers, symmetric powers, exterior powers, duals, adjoints
  • Automorphisms of vector spaces (row reduction and generators and relations for GLn),
  • ℤ-modules (Smith normal form and the fundamental theorem of abelian groups),
  • ℂ[t]-modules (eigenvectors, Rational canonical and Jordan normal forms),
  • vector spaces with bilinear form (Gram-Schmidt, Sylvester's theorem)

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 (no-one 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?).

  • The notation |G| doesn't match with the category of sets, where one can use Card(G). This is an example of "mixing categories". This notational mishap gets exacerbated with the notation |G:H| which means Card(G/H).
  • The inappropriate definition of a subgroup as a subset that satisfies: if h1, h2∈H then TRY MATH JAX HERE obfuscates the structure and is counterproductive.
  • For some time I have felt that rings are unhelpfully named and that ℤ-algebra would be much more sensible. I've rewritten the rings section to try to stimulate some shift in terminolgy in this direction. It is pleasant that both the French words algèbre and anneaux start with A, making it natural to comfortably denote a ℤ-algebra as A.

Thoughts and advice

  • The expectation is that the final exam will be 3 hours, Zoom supervised, no additional notes allowed. If this changes we will make an announcement of the change. Unless you can find some announcement changing this plan on this webpage and on Canvas, prepare for the case that no additional notes will be allowed on the exam.
  • Just as a carpenter without a hammer is mostly useless, a mathematician without a good place to work is mostly useless.
  • Tips for time management:
    • It is much easier (and safer) to run 45 min per day to attain 6 hours in a week and 24 hours in 4 weeks, than to run for 24 hours solid every fourth week on Sunday.
    • To actually run 45 min, it takes me at least 15 min to psyche myself up and convince myself that it is actually not raining and so therefore I should go running, and after a 45 min run I always walk for 5 min and I always go home and have a glass of milk and tell my wife (at length) how cool I am for running 45 min per day. All in all, I waste a good 40 min when I go running for 45 min. If I were more efficient (and every so often, but rarely, I am) then it would only takes me 50 min.
    • Measurement of time is a tricky thing and requires real discipline. Teaching and research faculty at University of Melbourne recently had to complete a survey on distribution of their time on the various activities of the job: Do I count the 6 times I had to go check my email and the weather and my iPhone in the time that I spend preparing my Group Theory and Linear Algebra lecture?
  • Tips for exam preparation:
    • The time that a 100m olympic runner (who wins a medal) is actually competing at the olympics is say (5 heats, 7sec each) 40 seconds. Successful performance during these 40 sec is impossible without adequate preparation.
    • The time that a Group Theory and Linear Algebra student spends on the final exam is 3 hours. Sucessful performance during these 3hours is .....
  • Prof. Ram reads email but generally does not respond by email. Usually these are collated and reponses to email queries are provided in the first few minutes of lectures. That way all students can benefit from the answer to the query.

Feedback

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.

Resources part I: recommended Readings

  • The 2019 and 2018 notes, available from the links at the bottom of the Home page for the course on Canvas. https://canvas.lms.unimelb.edu.au/courses/87189
  • The notes at http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html In particular, the following are particularly relevant to this course:
    • Lecture 1: Numbers
    • Numbers again
    • The integers ℤ
    • Z/nZ
    • The Rationals ℚ
    • The quaternions
    • Finite fields
    • Matrices Mn(F)
      • Numbers - Exercises
      • Z, Q, R, C - Exercises
    • Polynomials F[x], F[[x]], F(x) and F((x)); xn
    • The rings Q[[x]] and partial fractions
    • Sets, functions, cardinality and composition of functions
    • Relations
    • Ordered sets
    • Sets and functions proofs
      • Sets: Examples
      • Functions: Examples
      • Sets: Exercises
      • Functions: Exercises
      • Cardinality: Exercises
      • Ordered sets: Exercises
      • Orders: Exercises
      • Orders on Z,Q, R,C: Exercises
    • Fields
    • Vector spaces
    • Rings
    • Modules
    • Operations
      • Operations - Exercises
    • Groups, rings and fields
      • Groups and monoids - Exercises
    • Fields and integral domains (text)
    • Fields and integral domains (proofs)
    • Fields of fractions
    • Fields and ordered fields
    • Ordered fields
    • Absolute value
      • The triangle and Cauchy Schwartz inequalities
      • Absolute value - Exercises
    • Fields with valuation
    • Euclidean domains, Principal ideal domains and Unique factorization domains (text)
    • Euclidean domains, Principal ideal domains and Unique factorization domains (proofs)
    • Polynomial rings (text)
    • Polynomial rings (proofs)
  • Bhattacharya, Jain and Nagpaul, Basic Abstract Algebra, Cambridge University Press, 1994.
  • M. Artin, Algebra, Prentice-Hall 1991.
  • Wikipedia

The following problems page may have helpful examples:

  • http://www.exampleproblems.com/wiki/index.php/Main_Page