This course continues on from MATH1115, providing an in-depth development of fundamental concepts of calculus and linear algebra, with a particular emphasis on the underlying foundations of mathematics. The use and understanding of proof and abstract ideas will allow students to develop analytical skills which will form a base for further study in fundamental mathematics, as well as providing a foundation for a wide range of quantitative areas such as actuarial studies, computer science, economics, engineering, physics and statistics.

Topics to be covered include:

Calculus/Analysis - short introduction to metric spaces in the context of the calculus of functions of several variables, generalisation of the real analysis theory studied in MATH1115 to multivariable functions including limits and continuity, double integrals, Fubini's theorem, integrability of continuous functions, partial derivatives, gradients and directional derivatives, differentiation of multivariable functions, extreme values, vector functions, curves and parametrisations, infinite series, convergence tests, power series, Taylor series;

Linear Algebra – subspaces, span, linear independence, bases and dimension, linear maps, duality, eigenvalues and eigenvectors, inner product spaces, Gram-Schmidt orthogonalisation, operators on inner product spaces, the spectral theorem in finite dimensions, singular value decomposition.

Note: This is an Honours Pathway Course. It involves extra material and emphasises the use and understanding of proof and abstract ideas to a deeper conceptual level than MATH1014.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

1. Explain the fundamental concepts of analysis and linear algebra and their role in modern mathematics and applied contexts.
2. Demonstrate accurate and efficient use of analysis and linear algebra techniques.
3. Demonstrate capacity for mathematical reasoning through analysing, proving and explaining concepts and theorems from analysis and linear algebra.
4. Apply problem-solving using analysis and linear algebra techniques applied to diverse situations in physics, engineering and other mathematical contexts.

Examination Material or equipment

Information about examination material will be made available through the Examinations timetable.

Recommended Resources

Recommended textbooks include:

• Stewart, Multivariable Calculus
• Spivak, Calculus
• Hoffman and Kunze, Linear Algebra
• Axler, Linear Algebra Done Right (Also available as an eBook from Springer.)

There are a variety of online platforms you will use to participate in your study program as a whole at ANU. These could include videos for lectures and other instruction, two-way video conferencing for interactive learning, email and other messaging tools for communication, interactive web apps for formative and collaborative activities, print and/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

Staff Feedback

Students will be given feedback in the following forms in this course:

• Written comments on assignments.
• Verbal comments on group work in workshops.
• Lecturers and demonstrators may also give feedback to the whole class, to groups, to individuals, to focus groups.

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Other Information

Regarding collaboration when attempting assignments: you are encouraged to discuss the course material with your classmates as an aid to learning. However, every student is responsible for writing up their own solutions. Although you may work with others to understand the ingredients in a correct solution, producing the written solutions should be an individual effort. If you work with other students, please acknowledge this collaboration on the first page of your assignment. For example, write, “I discussed Problem 1 with Jane Doe and Problems 3 and 4 with John Doe.” When completing an assignment, you should consult only the textbooks, notes, lecturers, demonstrators, and classmates, as using other sources often compromises the learning goals of the assignment. However, if you do use any non-human resources (e.g., Wikipedia, other texts, online resources) besides the course textbooks and notes, you should cite these similarly.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Real analysis content to be covered: Infinite Series of Real Numbers; Power Series , Limits in R^n and Continuity of multivariable functions; Differential Calculus of multivariable Functions; iterated integrals and introductory multivariable integration.	Understanding of the content will be tested regularly and formatively via the weekly assignments and via the summative exams. Course activities will be in-person on campus (with remote adjustments only for participants with unavoidable travel restrictions/visa delays).
2	Linear algebra content to be covered (time permitting):Vector Spaces (mostly review from MATH1115: Definition of Vector Space, Subspaces, Span and Linear Independence, Bases, Dimension); Linear Maps (The Vector Space of Linear Maps, Null Spaces and Ranges, Representing a Linear Map by a Matrix, Invertibility and Isomorphic Vector Spaces, Duality); Eigenvalues, Eigenvectors, and Invariant Subspaces (Invariant Subspaces, Eigenvectors and Upper-Triangular Matrices, Eigenspaces and Diagonal Matrices); Inner Product Spaces (Inner Products and Norms, Orthonormal Bases, Orthogonal Complements and Minimization Problems); Operators on Inner Product Spaces (Self-Adjoint and Normal Operators, The Spectral Theorem, Positive Operators and Isometries, Polar Decomposition and Singular Value Decomposition).	As above.

Tutorial Registration

Workshops begin in Week 2. Please select a workshop session via MyTimetable. ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.

Assessment Summary

Assessment task	Value	Learning Outcomes
Weekly Assignments	30 %	1,2,3,4
Quizzes	20 %	1,2,3,4
End of Semester Examination	50 %	1,2,3,4

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the ANU Online website Students may choose not to submit assessment items through Turnitin. In this instance you will be required to submit, alongside the assessment item itself, hard copies of all references included in the assessment item.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Participation

In-person lectures and workshops are held on the main Acton campus of ANU. Students are expected to attend in-person; recordings of lectures will be made available through the Echo360 system where possible.

Examination(s)

Students should consult the course Wattle site and the ANU final examination timetable to confirm the date, time and mode/venue for the exam.

Assessment Task 1

Weekly Assignments

Weekly assignments will be submitted online via PDF upload to the Gradescope system and will be due from the end of Week 2 onwards, most Friday afternoons at 6pm Canberra time. A student’s lowest Assignment score will be discounted, and the average of their remaining Assignment scores will constitute 30% of their overall grade for MATH1116. This is an advanced stream course, so most of your written work will be formal proofs. Writing clear, concise, and compelling arguments is a skill that takes time to master, and there are a variety of resources posted on Wattle to help you. The proofs in the textbooks / notes, and in the posted solutions also provide excellent examples to study. Language is a mathematician’s primary tool; we don’t generally get to run experiments or do fieldwork, so in the absence of data to support our hypotheses, our arguments need to be sufficiently convincing.

It is intended that the marked assignments will be returned within 12 days after submission. Further details can be found on the course Wattle site.

Assessment Task 2

Quizzes

There will be at least 8 quizzes throughout the semester, during workshops. Each quiz will be marked out of 4 marks. A student's total marks from their quizzes will be added together, and the lesser of that total or the maximum cap of 20, will be be the mark (out of 20) that contributes 20% towards their overall course grade.

Assessment Task 3

End of Semester Examination

An end of semester examination is included in the assessment. The examination will be held during the university's official examination period for the semester. Students should consult the course Wattle site and the ANU final examination timetable to confirm the date, time and mode/venue for the exam.

Academic Integrity

Academic integrity is a core part of our culture as a community of scholars. At its heart, academic integrity is about behaving ethically. This means that all members of the community commit to honest and responsible scholarly practice and to upholding these values with respect and fairness. The Australian National University commits to embedding the values of academic integrity in our teaching and learning. We ensure that all members of our community understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with. The University has policies and procedures in place to promote academic integrity and manage academic misconduct. Visit the following Academic honesty & plagiarism website for more information about academic integrity and what the ANU considers academic misconduct. The ANU offers a number of services to assist students with their assignments, examinations, and other learning activities. The Academic Skills and Learning Centre offers a number of workshops and seminars that you may find useful for your studies.

Online Submission

You will be required to electronically make a declaration concerning academic integrity, as part of the submission of your assignments. Please keep a copy of the assignments for your records. MATH1116 does not use Turnitin, having been granted an exemption: submission of assignments is via Gradescope.

Hardcopy Submission

All assignment submission is electronic.

Late Submission

Individual assessment tasks may or may not allow for late submission. The policy regarding late submission is detailed below:

• Show Working: Late submission of the show working component (i.e., the portion submitted via Gradescope) of the assignments without an extension is penalised at the rate of 5% of the possible marks available per working day or part thereof.

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

Assignments will be returned electronically.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure The Course Convener may grant extensions for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Resubmission of assignments is not permitted in MATH1116, except that it is possible to update an early submission of an assignment up until its due date and time.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information. In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service — including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy. If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes. Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students