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

Unit code	MATH2022
Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prohibitions ?	MATH2922 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921)
Prerequisites ?	MATH1XX2 or (a mark of 65 or above in MATH1014)
Corequisites ?	None
Assumed knowledge ?	None
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Jonathan Spreer, jonathan.spreer@sydney.edu.au

Type	Description	Weight	Due	Length
Supervised exam ?	Final exam Multiple-choice and short answer questions	60%	Formal exam period	2 hours
Outcomes assessed:
Small test	Quiz 1 Multiple choice and short answer	10%	Week 05 Due date: 23 Mar 2023 at 23:59 Closing date: 23 Mar 2023	40 minutes
Outcomes assessed:
Assignment	Assignment 1 Extended written answers	5%	Week 07 Due date: 09 Apr 2023 at 23:59 Closing date: 16 Apr 2023	n/a
Outcomes assessed:
Small test	Quiz 2 Multiple choice and short answer	10%	Week 09 Due date: 27 Apr 2023 at 23:59 Closing date: 27 Apr 2023	40 minutes
Outcomes assessed:
Assignment	Assignment 2 Extended written answers	5%	Week 11 Due date: 14 May 2023 at 23:59 Closing date: 21 May 2023	n/a
Outcomes assessed:
Online task	Quiz 3 Multiple choice and short answer	10%	Week 13 Due date: 25 May 2023 at 23:59 Closing date: 25 May 2023	40 minutes
Outcomes assessed:

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, via Canvas 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 7 days past the original due date. Further extensions past this time will not be permitted.

Quizzes: Three quizzes will be held on the days of the compulsory practice classes (Weeks 5, 9, and 13). Quizzes will be online on Canvas, 40 minutes long, and will consist of multiple-choice questions and short answer questions.

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 sydney.edu.au/students/guide-to-grades.

For more information see 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.

Academic integrity

The Current Student website provides information on academic integrity 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 integrity breaches seriously.

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

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

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.

Weekly schedule

WK	Topic	Learning activity
Week 01	Revision of matrix arithmetic in the context of general fields	Lecture and tutorial (4 hr)
Week 01	Revision of matrix arithmetic in the context of general fields	Workshop (1 hr)
Week 02	1. Gaussian elimination in the context of general fields; 2. Permutations and cycle notation	Lecture and tutorial (4 hr)
Week 02	1. Gaussian elimination in the context of general fields; 2. Permutations and cycle notation	Workshop (1 hr)
Week 03	Elementary matrices, determinants, conjugation, even and odd permutations	Lecture and tutorial (4 hr)
Week 03	Elementary matrices, determinants, conjugation, even and odd permutations	Workshop (1 hr)
Week 04	Eigentheory, characteristic polynomials, Cayley-Hamilton theorem and dihedral groups	Lecture and tutorial (4 hr)
Week 04	Eigentheory, characteristic polynomials, Cayley-Hamilton theorem and dihedral groups	Workshop (1 hr)
Week 05	Diagonalisation, stochastic matrices, Perron’s theorem, subgroups and Cayley’s theorem	Lecture and tutorial (4 hr)
Week 05	Diagonalisation, stochastic matrices, Perron’s theorem, subgroups and Cayley’s theorem	Workshop (1 hr)
Week 06	Linear transformations of the plane, product decompositions of groups and vector spaces	Lecture and tutorial (4 hr)
Week 06	Linear transformations of the plane, product decompositions of groups and vector spaces	Workshop (1 hr)
Week 07	Basis and dimension, rank-nullity theorem for matrices and Lagrange’s theorem for groups	Lecture and tutorial (4 hr)
Week 07	Basis and dimension, rank-nullity theorem for matrices and Lagrange’s theorem for groups	Workshop (1 hr)
Week 08	Linear transformations, operators and matrix representations and rank-nullity theorem	Lecture and tutorial (4 hr)
Week 08	Linear transformations, operators and matrix representations and rank-nullity theorem	Workshop (1 hr)
Week 09	Jordan forms, group homomomorphisms and fundamental homomorphism theorem	Lecture and tutorial (4 hr)
Week 09	Jordan forms, group homomomorphisms and fundamental homomorphism theorem	Workshop (1 hr)
Week 10	Inner product spaces, orthogonality, projections and direct sum decompositions	Lecture and tutorial (4 hr)
Week 10	Inner product spaces, orthogonality, projections and direct sum decompositions	Workshop (1 hr)
Week 11	Gram-Schmidt process, QR-factorisations, adjoints and spectral theorems	Lecture and tutorial (4 hr)
Week 11	Gram-Schmidt process, QR-factorisations, adjoints and spectral theorems	Workshop (1 hr)
Week 12	Quadratic forms, diagonalisation matrix exponentials and applications	Lecture and tutorial (4 hr)
Week 12	Quadratic forms, diagonalisation matrix exponentials and applications	Workshop (1 hr)
Week 13	Spectral radius of a matrix, geometric series and revision	Lecture and tutorial (4 hr)
Week 13	Spectral radius of a matrix, geometric series and revision	Workshop (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

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.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

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

Media

Student links

About us

Connect

Current students

MATH2022: Linear and Abstract Algebra

Semester 1, 2023 [Normal day] - Remote

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

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

Media

Student links

About us

Connect