Outline

Overview

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 investigates and explores properties of linear functions, 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 group theory.

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	MATH1061 or MATH1961 or MATH1971 or MATH1X02 or (a mark of 65 or above in MATH1014)
Corequisites ?	None
Prohibitions ?	MATH2922 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921)
Assumed knowledge ?	None
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Sam Jeralds, samuel.jeralds@sydney.edu.au

Assessment

The census date for this unit availability is 31 March 2026

Details
Criteria

Type	Description	Weight	Due	Length	Use of AI
Written exam	Final exam Multiple-choice, short and/or extended answer questions	60%	Formal exam period	2 hours	AI prohibited
Outcomes assessed:
Contribution	Tutorial participation Collaborative problem solving	2%	Multiple weeks	1 hr	AI allowed
Outcomes assessed:
Out-of-class quiz	Early Feedback Task Multiple choice	2%	Week 03 Due date: 13 Mar 2026 at 23:59 Closing date: 13 Mar 2026	40 minutes	AI allowed
Outcomes assessed:
Written work	Assignment 1 Extended written answers	6%	Week 05 Due date: 27 Mar 2026 at 23:59 Closing date: 06 Apr 2026	3-5 pages	AI allowed
Outcomes assessed:
In-person written or creative task	Quiz 1 multiple choice	12%	Week 06 Due date: 02 Apr 2026 at 15:00	40 minutes	AI prohibited
Outcomes assessed:
Written work	Assignment 2 Extended written answers	6%	Week 11 Due date: 15 May 2026 at 23:59 Closing date: 25 May 2026	3-5 pages	AI allowed
Outcomes assessed:
In-person written or creative task	Quiz 2 multiple choice	12%	Week 12 Due date: 21 May 2026 at 15:00	40 minutes	AI prohibited
Outcomes assessed:
= early feedback task ?

Early feedback task

This unit includes an early feedback task, designed to give you feedback prior to the census date for this unit. Details are provided in the Canvas site and your result will be recorded in your Marks page. It is important that you actively engage with this task so that the University can support you to be successful in this unit.

Assessment summary

Detailed information for each assessment can be found on Canvas.

Assignments: There are two assignments. Each must be submitted electronically, as one single typeset or scanned PDF file only, by the deadline. Note that your assignment will not be marked if it is illegible or if it is submitted sideways or upside down. It is your responsibility to check that your assignment has been submitted correctly and that it is complete (check that you can view each page). Late submisions will receive a penalty. A mark of zero will be awarded for all submissions more than 10 days past the original due date. Further extensions past this time will not be permitted.

Early Feedback Task: This is an online quiz held in Week 3.

Quizzes: Two quizzes will be held in person at the time of the seminar in Weeks 6 and 12. You must sit the quiz at the time and location that appears as Assessment on your timetable. If you are unable to sit the quiz at that time for a valid reason, then you have the option to apply for Special Consideration or Special Arrangements.

Tutorial Participation: This is a satisfactory/non-satisfactory mark assessing whether or not you participate in class activities during the tutorials. It is 0.25 marks per tutorial class up to 8 tutorials (there are 12 tutorials in total).

Final Exam: There is one examination during the examination period at the end of Semester. Further information about the exam will be made available at a later date on Canvas. If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

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	Representing complete or close to complete mastery of the material.
Distinction	75 - 84	Representing excellence, but substantially less than complete mastery.
Credit	65 - 74	Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.
Pass	50 - 64	Representing 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 guide to grades.

Use of generative artificial intelligence (AI)

You can use generative AI tools for open assessments. Restrictions on AI use apply to secure, supervised assessments used to confirm if students have met specific learning outcomes.

Refer to the assessment table above to see if AI is allowed, for assessments in this unit and check Canvas for full instructions on assessment tasks and AI use.

If you use AI, you must always acknowledge it. Misusing AI may lead to a breach of the Academic Integrity Policy.

Visit the Current Students website for more information on AI in assessments, including details on how to acknowledge its use.

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.

Academic integrity

The University expects students to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

Our website provides information on academic integrity and the resources available to all students. This includes advice on how to avoid common breaches of academic integrity. Ensure that you have completed the Academic Honesty Education Module (AHEM) which is mandatory for all commencing coursework students

Penalties for serious breaches can significantly impact your studies and your career after graduation. It is important that you speak with your unit coordinator if you need help with completing assessments.

Visit the Current Students website for more information on AI in assessments, including details on how to acknowledge its use.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Support for students

The Support for Students Policy reflects the University’s commitment to supporting students in their academic journey and making the University safe for students. It is important that you read and understand this policy so that you are familiar with the range of support services available to you and understand how to engage with them.

The University uses email as its primary source of communication with students who need support under the Support for Students Policy. Make sure you check your University email regularly and respond to any communications received from the University.

Learning resources and detailed information about weekly assessment and learning activities can be accessed via Canvas. It is essential that you visit your unit of study Canvas site to ensure you are up to date with all of your tasks.

If you are having difficulties completing your studies, or are feeling unsure about your progress, we are here to help. You can access the support services offered by the University at any time:

Support and Services (including health and wellbeing services, financial support and learning support)
Course planning and administration
Meet with an Academic Adviser

Weekly schedule

WK	Topic	Learning activity
Week 01	Modular arithmetic, groups and fields	Lecture (3 hr)
Week 01	Modular arithmetic, groups and fields	Seminar (1 hr)
Week 02	Groups of symmetries and permutations	Lecture (3 hr)
	Modular arithmetic, groups and fields	Tutorial (1 hr)
	Modular arithmetic, groups and fields	Seminar (1 hr)
Week 03	Conjugation in symmetric groups, products of groups, group isomorphisms	Lecture (3 hr)
	Conjugation in symmetric groups, products of groups, group isomorphisms	Tutorial (1 hr)
	Conjugation in symmetric groups, products of groups, group isomorphisms	Seminar (1 hr)
Week 04	Groups wrap up, matrix arithmetic over fields, systems of equations	Lecture (3 hr)
	Groups wrap up, matrix arithmetic over fields, systems of equations	Tutorial (1 hr)
	Groups wrap up, matrix arithmetic over fields, systems of equations	Seminar (1 hr)
Week 05	Elementary matrices, invertibility and determinants, rotation and reflection matrices	Lecture (3 hr)
	Elementary matrices, invertibility and determinants, rotation and reflection matrices	Tutorial (1 hr)
	Elementary matrices, invertibility and determinants, rotation and reflection matrices	Seminar (1 hr)
Week 06	Eigenvalues and eigenvectors, characteristic polynomials, Cayley-Hamilton theorem	Lecture (3 hr)
	Eigenvalues and eigenvectors, characteristic polynomials, Cayley-Hamilton theorem	Tutorial (1 hr)
	Eigenvalues and eigenvectors, characteristic polynomials, Cayley-Hamilton theorem	Seminar (1 hr)
Week 07	Diagonalization and applications of eigenvalues/eigenvectors	Lecture (3 hr)
	Diagonalization and applications of eigenvalues/eigenvectors	Tutorial (1 hr)
	Diagonalization and applications of eigenvalues/eigenvectors	Seminar (1 hr)
Week 08	Cartesian space, linear transformations, compositions of maps	Lecture (3 hr)
Week 08	Cartesian space, linear transformations, compositions of maps	Seminar (1 hr)
Week 09	Abstract vector spaces, subspaces, linear independence	Lecture (3 hr)
	Abstract vector spaces, subspaces, linear independence	Tutorial (1 hr)
	Abstract vector spaces, subspaces, linear independence	Seminar (1 hr)
Week 10	Basis and dimension, coordinate systems, rank-nullity	Lecture (3 hr)
	Basis and dimension, coordinate systems, rank-nullity	Tutorial (1 hr)
	Basis and dimension, coordinate systems, rank-nullity	Seminar (1 hr)
Week 11	Linear transformations, representation by matrix, change of basis, matrix exponential	Lecture (3 hr)
	Linear transformations, representation by matrix, change of basis, matrix exponential	Tutorial (1 hr)
	Linear transformations, representation by matrix, change of basis, matrix exponential	Seminar (1 hr)
Week 12	Inner product spaces, projections, orthonormal bases	Lecture (3 hr)
	Inner product spaces, projections, orthonormal bases	Seminar (1 hr)
	Inner product spaces, projections, orthonormal bases	Tutorial (1 hr)
Week 13	Gram-Schmidt algorithm, Jordan canonical form, additional topics, revision	Lecture (3 hr)
	Gram-Schmidt algorithm, Jordan canonical form, additional topics, revision	Tutorial (1 hr)
	Gram-Schmidt algorithm, Jordan canonical form, additional topics, revision	Seminar (1 hr)

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.

Learning outcomes

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.

Outcomes
Graduate qualities

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

LO1. be fluent in analysing and constructing arguments involving matrix arithmetic, permutation and abstract groups, fields and vector spaces
LO2. understand the definitions, main theorems and corollaries for linearly independent sets, spanning sets, basis and dimension of vector spaces
LO3. be fluent with linear transformations and operators, and in interpreting, analysing and applying associated abstract phenomena using matrix representations and matrix arithmetic
LO4. develop appreciation and strong working knowledge of the theory and applications of elementary permutation groups, their decompositions and relationship to invertible phenomena in linear algebra
LO5. be fluent with important examples, theorems, algorithms and applications of the theory of inner product spaces, including processes and algorithms involving orthogonality, projections and optimisation.

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
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

We are continuing to improve the materials and resources for this unit, and thank students for their appreciative comments.

Additional information

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

Important: the University of Sydney regularly reviews units of study and reserves the right to change the units of study available annually. To stay up to date on available study options, including unit of study details and availability, refer to the relevant handbook.

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

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Early feedback task

Assessment summary

Assessment criteria

Use of generative artificial intelligence (AI)

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

On this page

Useful links

MATH2022: Linear and Abstract Algebra

Semester 1, 2026 [Normal day] - Camperdown/Darlington, Sydney

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Early feedback task

Assessment summary

Assessment criteria

Use of generative artificial intelligence (AI)

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

On this page

Useful links