
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 13:0014:00,
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.
Tutorials
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
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 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
Assignments
The homework assignments are at the following links:
(total 7% each = 5% for the first submission +2% for the second submission)
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) CayleyHamilton 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, orbitstabilizer 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 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 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
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:
𝔼^{2} and isometries
Tutorial sheets
 Tutorial sheet 1:
Greatest common divisors, Euclid's algorithm, arithmetic modulo m
 Tutorial sheet 2:
Fields, RSA cryptography
 Tutorial sheet 3:
Bases, linear transformations, eigenvalues, direct sums, invariant subspaces

Is S={(1,3), (3,4), (2,3)} in (F_{5})^{2} a basis?

Are {1, sin^{2} x, cos^{2}x} 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 Rlinear transformation

(F_{5})^{4} as a direct sum of invariant subspaces
,
Second part

Eigenvalues and eigenvectors of a 3cycle
 Tutorial sheet 4:
Minimal polynomials, diagonalisation
 Tutorial sheet 5:
CayleyHamilton theorem, Jordan normal form
 Tutorial sheet 6:
Properties and examples of groups; subgroups, cyclic groups, orders of elements
 Tutorial sheet 7:
Direct product, homomorphisms and isomorphisms, cosets
 Tutorial sheet 8:
Normal subgroups, Lagrange's theorem, quotient groups
 Tutorial sheet 9:
Inner products

Non inner products

Length of (12i, 2+3i) in ℂ^{2}

Find an orthonormal basis of ℂ^{2} containing (1+i,1i)

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 ℝ^{4}

W⊆ (W^{⊥})^{⊥}
and W=(W^{⊥})^{⊥} when dim(V) is finite
 Tutorial sheet 10:
Adjoints, spectral theorem
 Tutorial sheet 11:
Group actions, orbitstabiliser theorem, Sylow theorems
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
Student TODO list:
 Lecture 2, 6 August 2020: The clock number system
iPad chalkboard
Student TODO list:
 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
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
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/K7nbZltAoI
 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
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 Ainvariant subspaces
https://youtu.be/3WE5FRFa2XQ
 Determinant and characteristic polynomial of a diagonal matrix
 Diagonalisability corresponds to a basis of eigenvectors
 Minimal polynomials and The CayleyHamilton 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((λA)^{i})
Student TODO list:
 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
 ${\mathrm{GL}}_{n}(\mathbb{F})$
and its conjugation action on
${M}_{n}(\mathbb{F})$
 Definitions of group, group action, subgroup
 The examples
${\mathrm{GL}}_{n}(\mathbb{F})$,
${\mathrm{SL}}_{n}(\mathbb{F})$
and
${S}_{n}$
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 8889 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
 $G/\mathrm{ker}f\simeq \mathrm{im}f$.
 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(Stab_{G}(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 5Sylow subgroups
and the 2Sylow subgroups of D_{5} 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
 GramSchmidt
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:
 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:
 Principal ideal domains, gcd, unique factorisation
and partial fractions
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:
 The inner product on matrices
 Irreducible representations of SL_{2}(ℂ)
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:
 Proof of Jordan normal form
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:
 The other proof of Jordan normal form
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 notespdf 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:
CayleyHamilton theorem (pdf file)
 Lecture 15, 26 August 2011:
Inner products and GramSchmidt (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:
$G/\mathrm{ker}f\simeq \mathrm{im}f$ (pdf file)
 Lecture 25, 4 October 2011:
Group actions, orbits, stabilizers (pdf file)
 Lecture 26, 5 October 2011:
Centres and pgroups (pdf file)
 Lecture 27, 7 October 2011:
Proof of the OrbitStabilizer theorem (pdf file)
 Lecture 28, 11 October 2011:
The affine orthogonal group and isometries (pdf file)
 Lecture 29, 12 October 2011:
Isometries of E^{2} (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:
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 padic numbers ℤ_{p} and ℚ_{p}

The polynomial ring
$\u2102\left[\epsilon \right]$,
i.e.
the group algebra of ${\mathbb{Z}}_{\ge 0}$, and a study of its ideals

The ring
$\u2102[\epsilon ,{\epsilon}^{1}]$
i.e. the group algebra of $\mathbb{Z}$,
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:
 $\u2102$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 ${\mathrm{GL}}_{n}$),

$\mathbb{Z}$modules (Smith normal form and the
fundamental theorem of abelian groups),

$\u2102[\mathrm{t]}$modules
(eigenvectors, Rational canonical and Jordan normal forms),

vector spaces with bilinear form (GramSchmidt, 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 (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?).
 The notation $\leftG\right$ doesn't match with
the category of sets, where one can use $\mathrm{Card}\left(G\right)$. This is an example of "mixing categories". This notational
mishap gets exacerbated with the notation $G:H$ which means
$\mathrm{Card}(G/H)$.
 The inappropriate definition of a subgroup as a subset
that satisfies: if ${h}_{1},{h}_{2}\in H$
then TRY MATH JAX HERE obfuscates the structure and is counterproductive.
 For some time I have felt that rings are unhelpfully named and that
ℤalgebra 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 ℤalgebra 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 following problems page may have helpful examples: