Algebra 2

• Credit value: 30 credits at Level 5
• Convenor and tutor: Professor Sarah Hart
• Assessment: short, problem-based assignments (20%) and a three-hour examination (80%)

This module will give you a thorough grounding in the concepts and techniques of linear algebra, in a more general context than in first-year undergraduate modules. You’ll also gain an understanding of the key definitions and results of group theory. It will also prepare you for the Level 6 option Algebra 3.

Indicative module syllabus

Linear Algebra

• Fields, vector spaces, subspaces
• Linear independence, spanning sets, basis, the Steinitz exchange lemma and dimension
• Linear transformations and matrices
• Image, kernel and the rank-nullity formula
• Determinants of square matrices, inverses and Cramer’s rule
• Systems of linear equations and Gaussian elimination

Groups

• Binary operations, the properties of commutativity and associativity
• Identity elements and inverses
• Cayley tables
• Definition of a group
• Examples from geometry, permutations, matrices and number sets
• Homomorphisms and isomorphisms
• Cyclic groups and abelian groups
• Orders of elements and groups
• Subgroups and Lagrange’s Theorem
• Actions, G-sets, the orbit-stabilizer theorem, the orbit-counting lemma and applications such as colouring problems

By the end of this module, you will have:

• knowledge and understanding of, and the ability to use, mathematical techniques
• knowledge and understanding of a range of results in mathematics
• an appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments
• an understanding of the importance of assumptions and an awareness of where they are used and the possible consequences of their violation
• an appreciation of the power of generalisation and abstraction in the development of mathematical theories
• a deeper knowledge of some particular areas of mathematics, in particular, linear algebra and group theory.