Description:
 Aims: To introduce the abstract theories of groups and vector spaces and demonstrate their power through applications.
Learning Outcomes: After taking this unit the students should be able to:
* State and prove fundamental results of group theory and linear algebra.
* Demonstrate competence applying abstract ideas in specific examples.
* Compute determinants, eigenvalues and eigenvectors.
Skills: Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials)
Content: Axiomatic development of group theory.
Subgroups, homomorphisms, kernel and image.
Order of an element and of a group; cyclic groups. Isomorphism.
Permutations: cycle notation, sign of a permutation.
Actions, orbits and Cayley's theorem. Cosets, Lagrange's theorem and applications. OrbitStabilizer theorem and applications.
Axiomatic development of vector spaces.
Extending a linearly independent set to a basis. Dimension, rank, nullity. RankNullity theorem.
Determinants: definition, properties and computation. Adjugate and inverse formula. Invertibility of matrices. Cramer's rule.
Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicities. Diagonalisation.
