## PURE MTH 2106 - Algebra II

### North Terrace Campus - Semester 1 - 2020

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 Algebra II School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3.5 hours per week Y MATHS 1012 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

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.

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
• technology savvy
• professional and, where relevant, fully accredited
• forward thinking and well informed
• tested and validated by work based experiences
12
• 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. 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 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 Tutorials 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).
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
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

Due to the current COVID-19 situation modified arrangements have been made to assessments to facilitate remote learning and teaching. Assessment details provided here reflect recent updates.

A final exam, conducted using MyUni, worth 30% of your final mark, and to be held during the scheduled examination period;

A mid-semester test, conducted using MyUni, worth 20% of your final mark, and to be held on Thursday 30 April;

Fortnightly homework assignments, worth 25% of your final mark, distributed on MyUni as they have been on Wednesdays of even weeks and due on Fridays at 3pm of the following week;

Weekly quizzes testing the material of that week, worth 25% of your final mark, conducted using the Quiz facility on MyUni. These quizzes will be time-limited, but you will be able to attempt the questions at any time on the weekend of that week. The first
such quiz will be available this coming weekend (28-29 March), but it will not count at all: it will be used as an experiment to see how well the system works. But it should be of about the same length and level of difficulty that you can expect in general. Feedback will be very helpful.
##### 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.

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

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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