PURE MTH 7106 - Algebra

North Terrace Campus - Semester 1 - 2019

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 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
    Prerequisites MATHS 1012
    Assessment Ongoing assessment, exam
    Course Staff

    Course Coordinator: Associate Professor Nicholas Buchdahl

    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 what it means for a group to act on a set.
    7. Understand the concepts of vector space, linear transformation, isomorphism and related notions.
    8. Be able to construct and work with related objects: subspaces, sums, quotient spaces, dual spaces.
    9. Understand the notion of bilinear form.
    10. Understand the significance of Jordan canonical form.
    11. Be familiar with various methods of proof, including direct proof, constructive proof, proof by contradiction, induction.
    12. 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)
    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
    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
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
  • Learning Resources
    Required Resources
    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).
    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 relies on lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through the lectures. Written assignments aid the learning of the material and provide assessment opportunities for students to gauge their progress and understanding.

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

    Activity Quantity Workload Hours
    Lectures 36 90
    6 21
    Assignments 5 45
    Total 156
    Learning Activities Summary
    Lecture Outline
    1. Binary operations, groups, subgroups (2 lectures).
    2. Permutations, symmetric and alternating groups (2 lectures).
    3. Isomorphism of groups (1 lecture).
    4. Equivalence relations (1 lecture).
    5. Cosets and Lagrange's Theorem (2 lectures).
    6. Group homomorphisms (2 lectures).
    7. Normal subgroups and factor groups, simple groups, First Isomorphism Theorem (2 lectures).
    8. Groups acting on sets. Cauchy's theorem. (3 lectures).
    9. Symmetry and the dihedral groups. (2 lectures)
    10. Vector spaces, subspaces, linear independence, basis, dimension (3 lectures).
    11. Linear transformations. Sums and quotients of vector spaces. (3 lectures).
    12. Matrix with respect to basis, eigenvectors, similarity, dimension theorem (2 lectures).
    13. Linear functionals and the dual space, second dual space (1 lecture).
    14. Bilinear forms, congruent matrices, symmetric bilinear forms, quadratic forms (2 lectures).
    15. Inner products, norm, distance, orthogonality (2 lectures).
    16. Adjoints (1 lectures).
    17. Jordan canonical form (3 lectures).
    Tutorial Outline
    1. Tutorial 1: Groups.
    2. Tutorial 2: Permutations, isomorphism.
    3. Tutorial 3: Normal subgroups, quotient groups.
    4. Tutorial 4: Sums of spaces.
    5. Tutorial 5: Matrix of a linear transformation, linear functionals.
    6. Tutorial 6: Bilinear forms. Jordan canonical form.

    The lecture contents will be adjusted based on students' actual learning progress and outcome.
    Specific Course Requirements
    Small Group Discovery Experience
  • 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 17.5% all
    2.5% all
    Mid-term test 20% 1-5, 12
    Final Exam 60% all
    Assessment Related Requirements
    An aggregate score of at least 50% is required to pass the course.
    Assessment Detail
    Assessment Item  Distributed Due Date Weighting
    Assignment 1 week 2 week 3 3.5%
    Assignment 2 week 4 week 5 3.5%
    Assignment 3 week 6 week 7 3.5%
    Assignment 4 week 8 week 9 3.5%
    Assignment 5 week 10 week 11 3.5%
    Mid-term test week 6 week 6 20%
    Attendance at tutorials will count 2.5% towards the final mark for the course.
    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
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