Algebra
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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.
Objective
To introduce students to the basic ideas and methods of abstract algebra, with a focus on group theory and linear algebra. At the end of this course, students should be able to understand the concept of a group and a vector space, and be aware of examples of these structures in mathematics; appreciate and be able to prove and apply the basic results of group theory and linear algebra; understand the concepts of an equivalence relation and an isomorphism; understand the importance of proof, and appreciate the various different methods of proof encountered in the course.
Content
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.
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| Year |
Semester |
Level |
Units |
| 2012 |
1 |
2 |
3 |
Delivery
42 hours of lectures and tutorials
Assessment
Ongoing assessment 30%, exam 70%.
Graduate attributes
Linkage past
Prerequisite is Mathematics IB/IIM.
Linkage present
No present linkages have been noted.
Linkage future
This course is assumed knowledge for PURE MTH Groups and Rings III, for PURE MTH 3012 Fields and Geometry III, and for PURE MTH Coding and Cryptology.
Restrictions
None.
Recommended text
References: Fraleigh J.B.: A first course in abstract algebra (AddisonWesley)Lay D.C.: Linear algebra and its applications (Pearson)Lipschutz S.: Linear Algebra (Schaum's Outline Series)Katznelson Y. & Katznelson Y.R.: A (terse) introduction to linear algebra. (AMS)
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