618-202 Linear and Abstract Algebra

Note:

Credit Points:

Coordinator:

Prerequisite/s:

Timetable:

Contact:

Objectives:

On completion of this subject, students should:

Comprehend:

• basic and more advanced concepts of linear algebra such as vector spaces, inner product spaces, linear transformations, diagonalisation of matrices, Jordan canonical form and spectral decomposition;

• the basic algebraic structure of groups;

• the concepts of isomorphism, homomorphism and quotient algebraic structures.

Have developed:

• the ability to understand and write proofs of basic theorems;

• a knowledge of the principles and properties of abstract vector spaces and inner product spaces;

• skills in calculating eigenvalues and eigenvectors and in the diagonalisation of matrices;

• a knowledge of basic examples of the abstract algebraic structures;

• the ability to construct quotient objects.

Appreciate:

• the clarity and economy of abstract algebra, and the pleasure of proofs involving concepts rather than calculations.

Content:

Linear algebra: Revision and extension of basic concepts; vector spaces including complex spaces, inner products, linear transformations, eigenvalues and eigenvectors, dual spaces and the connection with inner products. The spectral theorem for normal matrices. Jordan normal form, without proof but with applications. Groups: abstract groups, examples including matrix groups and permutation groups; homomorphisms, normal subgroups, quotients and the first homomorphism theorem; group actions and permutation groups; conjugacy classes and their interpretation in symmetry groups, permutation groups and matrix groups. Applications, which may include: wallpaper groups, symmetry groups of regular polyhedra, permutation groups.

Assessment: