Algebra 1: Techniques and Applications

EMMS097S4 (30 credits)

Aims

This module aims to introduce the techniques of algebra and linear algebra together with some applications.

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings of lectures in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term.

Of the final course mark, 80% is based on a three-hour exam in June and the other 20% is from assessed coursework.

Coursework  will consist of short, problem based assignments. You will have around three weeks to complete each one.

The examination in June has two sections. Section A (worth 40%) consists of compulsory short questions. Section B (also worth 40%) contains several longer questions of which you must answer two.

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 & 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 & Eigenvectors: finding eigenvalues and eigenvectors of a square matrix, the characteristic equation, diagonalization 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 Outcomes

Subject Specific

Knowledge and understanding of, and the ability to use, mathematical and/or statistical methods and techniques.

In particular:

• Ability to combine mappings and permutations.
• Ability to solve systems of linear equations.
• Ability to find an orthogonal basis of a subspace of n-dimensional real space.
• Ability to evaluate the determinant, eigenvalues and eigenvectors of a square matrix.
• Knowledge when a square matrix is diagonalizable, and the ability to diagonalize such matrices.

Knowledge and understanding of a range of results in mathematics and/or statistics.

In particular:

• Knowledge of the basic notation and terminology of Set Theory.
• Knowledge of the properties of n-dimensional real space.
• Knowledge of the properties of standard functions of one variable.

Awareness of the use of mathematics and/or statistics to model problems in the natural and social sciences, and

• the ability to formulate such problems using appropriate notation.

In particular:

• Ability to encrypt and decrypt messages using simple cipher systems.
• Awareness of the limitations of certain cipher systems.
• Ability to model a finite stochastic process using a Markov matrix, and find the solution.
• Ability to model optimisation problems as a linear programme.

Ability to use a modern mathematical and/or statistical computer package with a programming facility, together with knowledge of other suitable packages.

In particular:

• Use a mathematical computer package to investigate and find solutions to the problems considered in the module.

Intellectual

• Ability to comprehend conceptual and abstract material.
• Develop a logical and systematic approach to problem solving.

Practical

• Ability to use a range of software packages including word processing and spreadsheets.
• Highly developed quantitative skills.
• Ability to transfer knowledge and expertise from one context to another.

Personal and Social

• Ability to learn independently using a variety of media.
• Time-management skills and organizational skills.
• General IT skills, including word processing and spreadsheets.
• Good communication skills, including the ability to write coherently.
• Ability to complete work in a limited time period.

Recommended books:

• Carol Whitehead, Guide² Abstract Algebra : this contains a good introduction to sets and some other topics covered in the first term.
• Anton, H and C Rorres, Elementary Linear Algebra with Applications, Wiley
• Kolman, B, Introductory Linear Algebra with Applications (6th edition), Prentice Hall
• Whitehead, C, Guide to Abstract Algebra (Macmillan Mathematical Guides), Macmillan