Unit of study_

# MATH2922: Linear and Abstract Algebra (Advanced)

## 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

Unit code MATH2922 Mathematics and Statistics Academic Operations 6 MATH2022 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921) MATH1902 or (a mark of 65 or above in MATH1002) None None Yes

### Teaching staff

Coordinator Zsuzsanna Dancso, zsuzsanna.dancso@sydney.edu.au Zsuzsanna Dancso

## Assessment

Type Description Weight Due Length
Supervised exam

Final 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
Due date: 17 Mar 2023 at 23:59
1 week
Outcomes assessed:
Assignment Individual Assignment
Written assignment submitted via canvas
15% Week 09
Due date: 27 Apr 2023 at 23:59
1 week
Outcomes assessed:
Tutorial quiz Webwork quizzes
Online quiz covering material from the previous weeks.
15% Weekly 90 minutes
Outcomes assessed:

### Assessment summary

• Final examination: At the end of semester there will be a final examin that testing the learning outcomes attained in lectures, tutorials and practice classes. 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.
• Online quizzes: You will 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 two assignments that will be available for download from the Online resources: the first is a group assignemnt, the second is individual. Groups for the first assignemnt will be set up from Week 2. Assignments must be submitted as PDF files via gradescope 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.

### 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.

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 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

There are no attendance marks in MATH2922, however regular class attendance and active participation has been a strong predictor of student success. Classes - including lectures - are interactive and include student work and collaboration.

In the remote stream, tutorials are live/hybrid (1 hr on Wednesday); lecture and practice class recordings are available through Canvas.

### 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.

All readings for this unit can be accessed through the Library, or 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.

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

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

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

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

Student feedback from previous years informed the current structure of assessments and learning activities.

### 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