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Algebra 1

Overview

  • Credit value: 30 credits at Level 4
  • Convenor and tutor: Dan Mcveagh
  • Assessment: three problem sets (10% each) and a three-hour examination (70%)

Module description

In this module we introduce you to the techniques of algebra and linear algebra together with some applications.

Indicative syllabus

  • Set Theory: subsets, power sets, complements, intersection, union and difference of two sets, Venn diagrams, partitions
  • Mappings: domain, codomain, range, injective, surjective and bijective mappings, composition of mappings, invertible mappings, induced mappings and restrictions
  • Permutations: composition of permutations, inverses, cycles notation, disjoint cycles, cycle decomposition, order of a permutation, transpositions, even and odd permutations
  • Elementary cryptography: crypyosystems, encryption and decryption, Caesar ciphers, substitution ciphers, transposition ciphers, attacks on cryptosytems
  • Matrices and systems of linear equations: operations on matrices, transposes, symmetric and antisymmetric matrices, invertible matrices, consistent and inconsistent equations, matrix form of a system of linear equations, elementary row operations, solving a system of linear equations, inverting a square matrix
  • Determinants: cofactors, evaluating the determinant of a square matrix, properties of the determinant
  • Real vectors: the dot product, the length of a vector, linear combinations, spanning subspaces, linearly independent vectors, bases, orthogonality, the angle between two vectors, orthogonal bases and the Gram-Schmidt process
  • Eigenvalues and eigenvectors: finding eigenvalues and eigenvectors of a square matrix, the characteristic equation, diagonalisation and powers of square matrices
  • Markov chains: transition matrices, state vectors, Markov matrices, regular transition matrices, steady state vectors
  • Linear programming: linear inequalities, formulation of a linear programme, objective function and constraints, graphical solutions, introduction to the simplex method

Learning objectives

By the end of this module, you will be able to:

  • combine mappings and permutations
  • solve systems of linear equations
  • find an orthogonal basis of a subspace of n-dimensional real space
  • evaluate the determinant, eigenvalues and eigenvectors of a square matrix
  • understand when a square matrix is diagonalisable, and diagonalise such matrices
  • understand the basic notation and terminology of Set Theory
  • understand the properties of n-dimensional real space and of standard functions of one variable
  • encrypt and decrypt messages using simple cipher systems
  • show awareness of the limitations of certain cipher systems
  • model a finite stochastic process using a Markov matrix, and find the solution
  • model optimisation problems as a linear program
  • use a mathematical computer package to investigate and find solutions to the problems considered in the module.