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Unit of study_

MATH2922: Linear and Abstract Algebra (Advanced)

Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit is an advanced version of MATH2022, with more emphasis on the underlying concepts and on mathematical rigour. This unit investigates and explores properties of vector spaces, matrices and linear transformations, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract groups theory. The unit culminates in studying inner spaces, quadratic forms and normal forms of matrices together with their applications to problems both in mathematics and in the sciences and engineering.

Details

Academic unit Mathematics and Statistics Academic Operations
Unit code MATH2922
Unit name Linear and Abstract Algebra (Advanced)
Session, year
? 
Semester 1, 2021
Attendance mode Normal day
Location Camperdown/Darlington, Sydney
Credit points 6

Enrolment rules

Prohibitions
? 
MATH2022 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921)
Prerequisites
? 
MATH1902 or (a mark of 65 or above in MATH1002)
Corequisites
? 
None
Available to study abroad and exchange students

Yes

Teaching staff and contact details

Coordinator Zsuzsanna Dancso, zsuzsanna.dancso@sydney.edu.au
Lecturer(s) Zsuzsanna Dancso , zsuzsanna.dancso@sydney.edu.au
Tutor(s) Diana Nguyen , diana.nguyen@sydney.edu.au
Type Description Weight Due Length
Final exam (Take-home short release) Type D final exam Final take-home exam
Final exam covering all of the material from lectures and tutorials.
60% Formal exam period 2 hours
Outcomes assessed: LO1 LO12 LO11 LO10 LO9 LO8 LO7 LO6 LO5 LO4 LO3 LO2
Assignment group assignment Group Assignment
Group written assignment
10% Week 04 1 week
Outcomes assessed: LO1 LO12 LO11
Assignment Assignment 1
Written assignment submitted via canvas
10% Week 08 n/a
Outcomes assessed: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO11 LO12
Assignment Assignment 2
Written assignment
10% Week 12 n/a
Outcomes assessed: LO7 LO8 LO9 LO10 LO11 LO12
Tutorial quiz Online quizzes
Online quiz covering material from the previous weeks.
10% Weekly 90 minutes
Outcomes assessed: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12
group assignment = group assignment ?
Type D final exam = Type D final exam ?
  • Final examination: At the end of semester there will be a take home examination that will test the learning outcomes attained in lectures, tutorials and practice classes. 
  • Online quizzes: You should complete an online quiz each week. The first online quiz will be due in week 2, and you will have a 6-day window in which to complete each quiz. The quizzes are timed and you have 90 minutes from the time you start; if you start a quiz within 90 minutes of the closing time, you will only get the remaining time to complete it (not the full 90 minutes). After the quiz has closed you will be able to review your answers and you will be given the solutions. Only your ten best quizzes count towards your final mark. There are no extensions.
  • Assignments: There are three assignments that will be available for download from the Online resources: the first is a group assignemnt, the later two are individual. Groups for the first assignemnt will be set up from Week 2. Assignments must be submitted as PDF files via Canvas before the deadline. All assignments are checked with TurnItIn, which is an internet-based plagiarism-prevention service that is designed to find similarities between all of the submitted assignments, including those from previous years, as well as checking for similarities with the literature.

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Complete or close to complete mastery of the material

Distinction

75 - 84

Excellence, but substantially less than complete mastery

Credit

65 - 74

A creditable performance that goes beyond routine knowledge and understanding, but less than excellence

Pass

50 - 64

At least routine knowledge and understanding over a spectrum of topics and important ideas
and concepts in the course

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see sydney.edu.au/students/guide-to-grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

  • Deduction of 5% of the maximum mark for each calendar day after the due date.
  • After ten calendar days late, a mark of zero will be awarded.

Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

Academic integrity

The Current Student website provides information on academic honesty, academic dishonesty, and the resources available to all students.

The University expects students and staff to act ethically and honestly and will treat all allegations of academic dishonesty or plagiarism seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic dishonesty. If such matches indicate evidence of plagiarism or other forms of dishonesty, your teacher is required to report your work for further investigation.

WK Topic Learning activity Learning outcomes
Week 01 Definition and basic properties of groups and group homomorphisms and subgroup Lecture and tutorial (4 hr)  
Definition and basic properties of groups and group homomorphisms and subgroup Seminar (1 hr)  
Week 02 Cosets, Lagrange’s theorem, symmetric and alternating groups and Cayley’s theorem Lecture and tutorial (4 hr)  
Cosets, Lagrange’s theorem, symmetric and alternating groups and Cayley’s theorem Seminar (1 hr)  
Week 03 Definition of fields, vector spaces, subspaces and spanning sets Lecture and tutorial (4 hr)  
Definition of fields, vector spaces, subspaces and spanning sets Seminar (1 hr)  
Week 04 Linear independence, bases, dimension and the replacement theorem Lecture and tutorial (4 hr)  
Linear independence, bases, dimension and the replacement theorem Seminar (1 hr)  
Week 05 Linear transformations, matrix representations and coordinate vectors Lecture and tutorial (4 hr)  
Linear transformations, matrix representations and coordinate vectors Seminar (1 hr)  
Week 06 Composition of linear transformations, Gaussian elimination and Rank-Nullity theorem Lecture and tutorial (4 hr)  
Composition of linear transformations, Gaussian elimination and Rank-Nullity theorem Seminar (1 hr)  
Week 07 Transition matrices and change of basis Lecture and tutorial (4 hr)  
Transition matrices and change of basis Seminar (1 hr)  
Week 08 Determinants, eigenvalues, eigenvectors and the characteristic polynomial Lecture and tutorial (4 hr)  
Determinants, eigenvalues, eigenvectors and the characteristic polynomial Seminar (1 hr)  
Week 09 Generalised eigenspaces and the decomposition theorem Lecture and tutorial (4 hr)  
Generalised eigenspaces and the decomposition theorem Seminar (1 hr)  
Week 10 Jordan normal form and the Cayley-Hamilton theorem and exponential of a matrix Lecture and tutorial (4 hr)  
Jordan normal form and the Cayley-Hamilton theorem and exponential of a matrix Seminar (1 hr)  
Week 11 Inner product spaces, Cauchy-Schwartz inequality and Gram-Schmidt orthogonalisation Lecture and tutorial (4 hr)  
Inner product spaces, Cauchy-Schwartz inequality and Gram-Schmidt orthogonalisation Seminar (1 hr)  
Week 12 Unitary and Hermitian matrices, QR-factorisation and least squares approximation Lecture and tutorial (4 hr)  
Unitary and Hermitian matrices, QR-factorisation and least squares approximation Seminar (1 hr)  
Week 13 Singular value decompositions, Frobenius-Perron theorem and revision Lecture and tutorial (4 hr)  
Singular value decompositions, Frobenius-Perron theorem and revision Seminar (1 hr)  

Attendance and class requirements

Due to the exceptional circumstances caused by the COVID-19 pandemic, attendance requirements for this unit of study have been amended. Where online tutorials/workshops/virtual laboratories have been scheduled, students should make every effort to attend and participate at the scheduled time. Penalties will not be applied if technical issues, etc. prevent attendance at a specific online class. In that case, students should discuss the problem with the coordinator, and attend another session, if available.

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Required readings

All readings for this unit can be accessed through the Library eReserve, available through Canvas. Course materials will be available for download from the unit of study web page. The WebWork quizzes, which are linked from the unit web page, need to be completed each week.

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

  • LO1. appreciate the basic concepts and problems of linear algebra and be able to apply linear algebra to solve problems in mathematics, science and engineering
  • LO2. understand the definitions of fields and vector spaces and be able to perform calculations in real and complex vector spaces, both algebraically and geometrically
  • LO3. determine if a system of equations is consistent and find its general solution
  • LO4. compute the rank of a matrix and understand how the rank of a matrix relates to the solution set of a corresponding system of linear equations
  • LO5. compute the eigenvalues, eigenvectors, minimal polynomials and normal forms for linear transformations
  • LO6. use the definition and properties of linear transformations and matrices of linear transformations and change of basis, including kernel, range and isomorphism
  • LO7. compute inner products and determine orthogonality on vector spaces, including Gram-Schmidt orthogonalisation
  • LO8. identify self-adjoint transformations and apply the spectral theorem and orthogonal decomposition of inner product spaces, and the Jordan canonical form, to solving systems of ordinary differential equations
  • LO9. calculate the exponential of a matrix and use it to solve a linear system of ordinary differential equations with constant coefficients
  • LO10. identify special properties of a matrix, such as symmetric of Hermitian, positive definite, etc., and use this information to facilitate the calculation of matrix characteristics
  • LO11. demonstrate accurate and efficient use of advanced algebraic techniques and the capacity for mathematical reasoning through analysing, proving and explaining concepts from advanced algebra
  • LO12. apply problem-solving using advanced algebraic techniques applied to diverse situations in physics, engineering and other mathematical contexts

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1 Depth of disciplinary expertise

Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.

GQ2 Critical thinking and problem solving

Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.

GQ3 Oral and written communication

Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.

GQ4 Information and digital literacy

Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.

GQ5 Inventiveness

Generating novel ideas and solutions.

GQ6 Cultural competence

Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.

GQ7 Interdisciplinary effectiveness

Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.

GQ8 Integrated professional, ethical, and personal identity

An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.

GQ9 Influence

Engaging others in a process, idea or vision.

Outcome map

Learning outcomes Graduate qualities
GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9
The tutorial and lecture summaries and online quizzes have been updated in response to constructive suggestions.

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

  • No eating or drinking is allowed in any laboratory under any circumstances
  • A laboratory coat and closed-toe shoes are mandatory
  • Follow safety instructions in your manual and posted in laboratories
  • In case of fire, follow instructions posted outside the laboratory door
  • First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
  • As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.