PURE MTH 2106 - Algebra II

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 in the physical sciences. 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. Homomorphism and isomorphism 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 2106
    Course Algebra II
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3.5 hours per week
    Available for Study Abroad and Exchange Y
    Prerequisites 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 in the physical sciences.

    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. Homomorphism and isomorphism 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

    No information currently available.

    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. Be able to represent a linear transformation by a matrix with respect to given ordered bases.
    9. Understand the significance of Jordan canonical form.
    10. Understand the notion of bilinear form and inner product.
    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.
    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
    11,12
    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
    Self-awareness and emotional intelligence
    • a capacity for self-reflection and a willingness to engage in self-appraisal
    • open to objective and constructive feedback from supervisors and peers
    • able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
    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).
    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.
    Workload

    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
    Tutorials 6 21
    Assignments 5 45
    Total 156
    Learning Activities Summary
    Lecture Outline
    1. Equivalence relations (1 lecture).
    2. Groups, subgroups, symmetries, cyclic groups (3 lectures).
    3. Permutations, symmetric and alternating groups (2 lectures).
    4. Isomorphism of groups (1 lecture).
    5. Cosets and Lagrange's Theorem (3 lectures).
    6. Group homomorphisms (2 lectures).
    7. Normal subgroups and factor groups, simple groups, First Isomorphism Theorem (3 lectures).
    8. Vector spaces, subspaces, linear independence, basis, dimension (3 lectures).
    9. Linear and direct sums of spaces, quotient spaces (2 lectures).
    10. Linear transformations (1 lecture).
    11. Matrix with respect to basis, eigenvectors, similarity, dimension theorem (2 lectures).
    12. Projections, invariant spaces (1 lecture).
    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 (3 lectures).
    16. Linear operators, adjoints (3 lectures).
    17. Jordan canonical form (2 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: Inner products.
    The lecture contents will be adjusted based on students' actual learning progress and outcome.
    Specific Course Requirements
    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
    Tutorials 2.5% all
    Mid-term test 22.5% 1-5, 11, 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%
    Assignment 2 week 4 week 5 3%
    Assignment 3 week 6 week 7 3%
    Assignment 4 week 8 week 9 3%
    Assignment 5 week 10 week 11 3%
    Mid-term test week 6 week 6 22.5%
    Attendance at tutorials will count 2.5% towards the final mark for the course.
    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
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