Arun Ram: Gp Thy and Lin Alg Sem II 2011

MAST20022
(620-297)
Group Theory and Linear Algebra

Semester II 2011

Lecturer: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au

Time and Location:
       Lecture Tuesday 10:00-11:00 Old Geology 1
       Lecture Wednesday 12:00-1:00 Old Geology 1
       Lecture Friday 2:15-3:15 Old Geology 1
       Practical Tuesday 11:00-12:00 Richard Berry G10
       Practical Wednesday 3:15-4:15 Richard Berry G10
       Practical Wednesday 11:00-12:00 Asia Centre 120
       Practical Thursday 2:15-3:15 Richard Berry G10
       Practical Friday 12:00-1:00 Richard Berry G4

Announcements

No books, notes, calculators, ipods, ipads, phones, etc at the exam.
Tips to avoid freaking out:
- The assignments are designed to take "an average of 6 hours per week". This is an average.
- The assignments can be reformatted to reduce the freak factor: See Assignment 1 (pdf file) from 2009 Real Analysis and Applications as an example.
- The assignments and the course are designed to make you know exactly what is on the exam, practice what is on the exam and do well on the exam.
- The assignments are worth 20% of the total mark. If you skip a few questions it will affect your total mark very little.
- Thousands of students have made it through this course format with Professor Ram in the past (and are proud to tell the tale). You can do it too.
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 .....
Consultation hours for Prof. Ram will be Mondays 3:45-5:45pm in Old Geology 1.
Prof. Ram reads email but generally does not respond.
The start of semester pack includes: Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), Beyond third year (pdf file), Vacation scholarships (pdf file), SSLC (pdf file).
It is University Policy that:
“a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results. Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).
Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).
Students must use UNICARD to print documents. The UNICARD printer is located near the G70 computer lab. For more information about printing at the University and for locations of UNICARD uploaders direct students to Student IT Support: http://www.studentit.unimelb.edu.au/printingandscanning/printing.html

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2011/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%).

The plagiarism declaration is available here. It is STRONGLY suggested that you turn in problems from the Problem sheets weekly (in tutorial).

The homework assignments will soon appear below:

Assignment 1: Due 23 August: Do problems from Problem sheets for weeks 1-4. Problems from sheet 1-4 will be accepted by your tutor anytime before 23 August. It is STRONGLY suggested that you turn in problems from the Problem sheets weekly (in tutorial). The marker will briefly look through your assignment and try to give you feedback on and a mark reflecting how you are progressing towards doing well on the final exam.
Assignment 2: Due 4 October: Do problems from Problem sheets for weeks 5-8. Problems from sheet 5-8 will be accepted by your tutor anytime between 23 August and 3 October. It is STRONGLY suggested that you turn in problems from the Problem sheets weekly (in tutorial). The marker will briefly look through your assignment and try to give you feedback on and a mark reflecting how well you are progressing towards doing well on the final exam.
Assignment 3: Due 1 November: Do problems from Problem sheets for weeks 9-12. Problems from sheet 9-12 will be accepted by your tutor anytime between 4 October and 28 October. It is STRONGLY suggested that you turn in problems from the Problem sheets weekly (in tutorial). The marker will briefly look through your assignment and try to give you feedback on how you are progressing towards doing well on the final exam.

Resources part I: recommended Texts

Craig Hodgson, 620-297 Notes. Get this from the LMS: The direct link is http://app.lms.unimelb.edu.au/bbcswebdav/courses/MAST20022/HTML/PUBLIC/111010Notes_297all.pdf
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 M_n(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

Resources part II: Lectures and lecture notes

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 2010: 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 2010: 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: $G / \ker f ≃ im f$ (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 2010: Revision: Working sample randomly chosen problems
Lecture 35, 26 October 2010: Revision: Maths, Music and the Weil conjectures
Lecture 36, 28 October 2010: Revision: C[t]-modules (pdf file)

Resources part III: Other notes

Various lecture notes from the past that will be useful and supplemented during the term.

Feedback

Every subject at the University of Melbourne uses a student questionnnaire to let teaching staff know what students think about the quality of teaching in that subject. This is now administered online near the end of the semster. As such, it is too late to affect the teaching for the cohort of students that answer the questionnaire.

Feedback to students based on 2009 questionnaires for Real Analysis:

The student survey last year showed high student satisfaction with the course. Most elements of last year's course are being retained.
Exam performance demonstrated that students had learned concepts and the general framework well, but were weak on skill (they knew what a hammer is for but were unable to use it to hammer in a nail effectively). Skill level is an important goal for this course and this semester there will be a determined effort to get the skill level of all students to a high level:
- The problem sheets will be very directed towards the final exam.