Course overview
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 learning outcomes
- Appreciate that common features of certain mathematical objects can be abstracted and studied.
- Understand equivalence relations and partitions.
- Understand the concepts of groups, group homomorphism and isomorphism and related notions.
- Be familiar with common examples of groups of both finite and infinite order.
- Be able to construct and work with related objects: subgroups, cartesian products, quotient groups.
- Understand the concepts of vector space, linear transformation, isomorphism and related notions.
- Be able to construct and work with related objects: subspaces, sums, quotient spaces, dual spaces.
- Understand the notion of bilinear form.
- Understand the significance of Jordan canonical form.
- Be familiar with various methods of proof, including direct proof, constructive proof, proof by contradiction, induction.
- Develop skills in creative and critical thinking, problem solving, logical writing and clear communication of mathematical ideas.