## PURE MTH 7106 - Algebra

### North Terrace Campus - Semester 1 - 2015

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 Algebra Pure Mathematics Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 4 hours per week Y MATHS 1012 ongoing assessment 30%, exam 70%
##### Course Staff

Course Coordinator: Dr Hang Wang

##### Course Timetable

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

• Learning Outcomes
##### Course Learning Outcomes
1. Understand equivalence relations and partitions.
2. Appreciate that common features of certain mathematical objects can be abstracted and studied as groups and vector spaces.
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, cartisian products, quotient groups.
6. Understand the concepts of vector space, linear transformation and isomorphism and related notions.
7. Be able to construct and work with related objects: subspaces, sums, quotient spaces, dual space.
8. Be able to represent a linear transformation by a matrix with respect to a given basis.
9. Understand the significance of Jordan canonical form.
10. Understand the notion of bilinear form and inner product.
11. Be familiar with various method of proof, including direct proof, constructive proof, proof by contradiction, induction.

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

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 2,5,7,11
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 2,5,7,11
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
• Learning Resources
None.
##### Recommended Resources
1. Fraleigh, J. B.: A first course in abstract algebra (Addison Wesley).
2. Durbin, J. R.: Modern algebra (Wiley).
3. Gallian, J. A.: Contemporary abstract algebra (Houghton Mifflin).
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 MyUni 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. A sequence of written assignments and two midterm exams provide the 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 92 Tutorials 6 24 Assignments 5 40 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

Tutorial 1: Groups.
Tutorial 2: Permutations, isomorphism.
Tutorial 3: Normal subgroups, quotient groups.
Tutorial 4: Sums of spaces.
Tutorial 5: Matrix of a linear transformation, linear functionals.
Tutorial 6: Inner products.
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 20% all Midterm Exams 10% all Final Exam 70% 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 4% Assignment 2 week 4 week 5 4% Assignment 3 week 6 week 7 4% Assignment 4 week 8 week 9 4% Assignment 5 week 10 week 11 4% Midterm Exam 1 week 6 week 6 5% Midterm Exam 2 week 10 week 10 5%
##### Submission
1. All written assignments are to be submitted at the designated lecture with a signed cover sheet attached.
2. Assignments will have a one week turn-around time for feedback to students.

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

M10 (Coursework Mark Scheme)
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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