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Algebra 2

Overview

  • 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%)

Module description

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

Learning objectives

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.