# Algebra 1

## Overview

• Credit value: 30 credits at Level 4
• Assessment: a three-hour examination in the summer term (80%) and coursework (20%)

## Module description

This module introduces the techniques of algebra and linear algebra together with some applications.

### Indicative module content

• 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
• demonstrate an 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 programme
• use a mathematical computer package to investigate and find solutions to the problems considered in the module.