## PURE MTH 2106 - Algebra II

### North Terrace Campus - Semester 1 - 2022

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: 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.

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

University Graduate Attribute Course Learning Outcome(s)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

all

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

all

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

12

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

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 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.

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 12 24 Assignments 5 40 Total 154
##### 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.
• 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 Quizzes 15% 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.

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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