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 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.

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	MATH1902 or (a mark of 65 or above in MATH1002)
Corequisites ?	None
Prohibitions ?	MATH2022 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	Zsuzsanna Dancso, zsuzsanna.dancso@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam (Take-home short release)	Final take-home exam Final exam covering all of the material from lectures and tutorials.	60%	Formal exam period	2 hours
Outcomes assessed:
Assignment	Group Assignment Group written assignment	10%	Week 04	1 week
Outcomes assessed:
Assignment	Assignment 1 Written assignment submitted via canvas	10%	Week 08	n/a
Outcomes assessed:
Assignment	Assignment 2 Written assignment	10%	Week 12	n/a
Outcomes assessed:
Tutorial quiz	Online quizzes Online quiz covering material from the previous weeks.	10%	Weekly	90 minutes
Outcomes assessed:
= group assignment ? = Type D final exam ?

Assessment summary

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 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.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

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	Definition and basic properties of groups and group homomorphisms and subgroup	Lecture and tutorial (4 hr)
Week 01	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)
Week 02	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)
Week 03	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)
Week 04	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)
Week 05	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)
Week 06	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)
Week 07	Transition matrices and change of basis	Seminar (1 hr)
Week 08	Determinants, eigenvalues, eigenvectors and the characteristic polynomial	Lecture and tutorial (4 hr)
Week 08	Determinants, eigenvalues, eigenvectors and the characteristic polynomial	Seminar (1 hr)
Week 09	Generalised eigenspaces and the decomposition theorem	Lecture and tutorial (4 hr)
Week 09	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)
Week 10	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)
Week 11	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)
Week 12	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)
Week 13	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

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. 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
Learning outcomes

Responding to student feedback

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

The tutorial and lecture summaries and online quizzes have been updated in response to constructive suggestions.

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

Attendance and class requirements

Study commitment

Required readings

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

MATH2922: Linear and Abstract Algebra (Advanced)

Semester 1, 2021 [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

Attendance and class requirements

Study commitment

Required readings

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