PURE MTH 7106 - Algebra

North Terrace Campus - Semester 1 - 2021

Knowledge of group theory and of linear algebra is important for an understanding of many areas of pure and applied mathematics, including advanced algebra and analysis, number theory, coding theory, cryptography and differential equations. There are also important applications to physics and chemistry. Topics covered are (1) Equivalence relations (2) Groups: subgroups, cyclic groups, cosets, Lagrange's theorem, normal subgroups and factor groups. Examples of finite and infinite groups, including groups of symmetries and permutations, groups of numbers and matrices. Homomorphisms and isomorphisms of groups. (3) Linear algebra: vector spaces, bases, linear transformations and matrices, subspaces, sums and quotients of spaces, dual spaces, bilinear forms and inner product spaces, and canonical forms.

  • General Course Information
    Course Details
    Course Code PURE MTH 7106
    Course Algebra
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    Contact Up to 4 hours per week
    Available for Study Abroad and Exchange Y
    Assumed Knowledge MATHS 1012
    Course Description Knowledge of group theory and of linear algebra is important for an understanding of many areas of pure and applied mathematics, including advanced algebra and analysis, number theory, coding theory, cryptography and differential equations. There are also important applications to physics and chemistry.

    Topics covered are (1) Equivalence relations (2) Groups: subgroups, cyclic groups, cosets, Lagrange's theorem, normal subgroups and factor groups. Examples of finite and infinite groups, including groups of symmetries and permutations, groups of numbers and matrices. Homomorphisms and isomorphisms of groups. (3) Linear algebra: vector spaces, bases, linear transformations and matrices, subspaces, sums and quotients of spaces, dual spaces, bilinear forms and inner product spaces, and canonical forms.
    Course Staff

    Course Coordinator: Dr Raymond Vozzo

    Course Timetable

    The full timetable of all activities for this course can be accessed from Course Planner.

  • Learning Outcomes
    Course Learning Outcomes
    1. Appreciate that common features of certain mathematical objects can be abstracted and studied.

    2. Understand equivalence relations and partitions.

    3. Understand the concepts of groups, group homomorphism and isomorphism and related notions.

    4. Be familiar with common examples of groups of both finite and infinite order.

    5. Be able to construct and work with related objects: subgroups, cartesian products, quotient groups.

    6. Understand the concepts of vector space, linear transformation, isomorphism and related notions.

    7. Be able to construct and work with related objects: subspaces, sums, quotient spaces, dual spaces.

    8. Understand the notion of bilinear form.

    9. Understand the significance of Jordan canonical form.

    10. Be familiar with various methods of proof, including direct proof, constructive proof, proof by contradiction, induction.

    11. Develop skills in creative and critical thinking, problem solving, logical writing and clear communication of mathematical ideas.
    University Graduate Attributes

    This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

    University Graduate Attribute Course Learning Outcome(s)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    all
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    all
    Teamwork and communication skills
    • developed from, with, and via the SGDE
    • honed through assessment and practice throughout the program of studies
    • encouraged and valued in all aspects of learning
    12
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    12
  • Learning Resources
    Required Resources
    None.
    Recommended Resources
    1. Fraleigh, J. B.: A first course in abstract algebra (Addison Wesley).
    2. Durbin, J. R.: Modern algebra (Wiley).
    3. Herstein, I. N.: Topics in Algebra (Wiley).
    4. Lay, D. C.: Linear algebra and its applications (Pearson).
    5. Lipschutz, S.: Linear algebra (Schaum's Outline Series).
    6. Katznelson, Y. & Katznelson, Y. R.: A (terse) introduction to linear algebra (AMS).
    7. Axler, S.: Linear Algebra Done Right (Springer).
    Online Learning
    This course uses Canvas for providing electronic resources, such as lecture notes, assignment papers, sample solutions, etc. It is recommended that students make appropriate use of these resourses.


  • Learning & Teaching Activities
    Learning & Teaching Modes
    This course runs in a "flipped classroom" model, whereby students are expected to watch the lecture topic videos and complete any necessary reading before coming to active learning classes (tutorials), which provide exercises and example problems to enhance the understanding obtained through the videos. Written assignments aid the learning of the material and provide assessment opportunities for students to gauge their progress and understanding.
    Workload

    The information below is provided as a guide to assist students in engaging appropriately with the course requirements.


    Activity Quantity Workload Hours
    Lecture videos 12 weeks 90
    Tutorials
    6 21
    Assignments 5 45
    Total 156
    Learning Activities Summary
    Topic Outline

    (1)Equivalence relations
    (2) Groups: subgroups, cyclic groups, cosets, Lagrange's theorem, normal subgroups and factor groups. Examples of finite and infinite groups, including groups of symmetries and permutations, groups of numbers and matrices. Homomorphisms and isomorphisms of groups.
    (3) Linear algebra: vector spaces, bases, linear transformations and matrices, subspaces, sums and quotients of spaces, dual spaces, bilinear forms and inner product spaces, and canonical forms.


    Specific Course Requirements
    None.
    Small Group Discovery Experience
    None.
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

    1. Assessment must encourage and reinforce learning.
    2. Assessment must enable robust and fair judgements about student performance.
    3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
    4. Assessment must maintain academic standards.

    Assessment Summary

    Component  Weighting Objective Assessed
    Assignments 15% all
    Quizzes
    15% all
    Tutorials/Workshops 5% all
    Mid-term test 15% 1-5, 10,11
    Final Exam 50% all
    Assessment Related Requirements
    An aggregate score of at least 50% is required to pass the course.
    Assessment Detail

    No information currently available.

    Submission
    Assignments will have a 2-week turn-around time for feedback to students.
    Course Grading

    Grades for your performance in this course will be awarded in accordance with the following scheme:

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    FNS   Fail No Submission
    F 1-49 Fail
    P 50-64 Pass
    C 65-74 Credit
    D 75-84 Distinction
    HD 85-100 High Distinction
    CN   Continuing
    NFE   No Formal Examination
    RP   Result Pending

    Further details of the grades/results can be obtained from Examinations.

    Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

    Final results for this course will be made available through Access Adelaide.

  • Student Feedback

    The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

    SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

  • Student Support
  • Policies & Guidelines
  • Fraud Awareness

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