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Semester 2

Honours Algebra (MATH10069)

Subject

Mathematics

College

SCE

Credits

20

Normal Year Taken

3

Delivery Session Year

2023/2024

Pre-requisites

Visiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling.

Course Summary

This is a core course for Honours Degrees involving Mathematics. It showcases the power of abstraction and brings together several different topics from earlier courses as well as new ideas, in order to present a view on some of the more advanced algebra that is critical for later courses and also in application, as well as of interest in its own right. The course also includes computer algebra, to support and illustrate some of the content.

Course Description

One great power of mathematics is abstraction. Here abstraction is the distillation of a mathematical idea to its essence - in the case of linear algebra, understanding what unifies points, lines and planes, and so on. This turns out to be very useful. Why? Well two apparently different ideas can turn out to have the same underlying structure, and then insight you get from one can reveal the other. Think of a dictionary, allowing you to translate the masterpieces of one language into another. [Of course, you still need to work hard to make a good translation.] Second, it is often easier to understand general structure than it is to understand specific examples. Many mathematicians think "if you can't answer a particular question, generalise it - it might become easier!". This course is about showcasing this, as well as giving you a solid base for more advanced topics in Year 4 and beyond. It also emphasises connections with other parts of mathematics, and will feature applications of the theory to problems, sometimes even beyond mathematics.The syllabus first covers abstract vector spaces and linear transformations. It then introduces rings and modules, their quotients, and the first isomorphism theorem. The multilinear algebra of determinants is studied, together with eigenvectors and eigenvalues, culminating in the Cayley-Hamilton theorem and the Perron-Frobenius Theorem. This is followed by an introduction to inner product spaces and the Spectral Theorem. The course then moves on to normal forms for linear transformations and particularly the Jordan Normal Form. Throughout the course, we will also work with a computer algebra system to learn about programming skills and data structures which are useful in Pure Mathematics and beyond. We will use these skills to investigate topics that are relevant to the theory being developed. Students will also carry out a group project which will require some computer algebra work.Linear Algebra1. Basic concepts in abstract linear algebra, abstract vector spaces, bases, linear maps, dimension, images and kernels.2. Linear transformations, choice of basis, Smith normal form. Rings and Modules1. Basic definitions and examples of rings, homomorphisms, kernels, images.2. Polynomials, their Euclidean algorithm, roots and algebraically closed fields.3. Basic definitions and examples of modules, homomorphisms, kernels, images.4. Quotient rings, modules and vector spaces; the first isomorphism theorem.Determinants and Eigenvalues1. Multilinear forms; characterisations of determinant.2. Eigenvalues and eigenvectors; diagonalisable and triangularisable linear mappings; Cayley-Hamilton Theorem.3. Perron-Frobenius Theorem and applications.Inner Product Spaces1. Basic definitions and examples of inner product spaces.2. Orthogonal projection; Gram-Schmidt.3. Adjoints of linear transformations; spectral theorem for finite dimensional inner product spaces.Jordan Normal Form1. The Jordan Normal Form.2. Applications of the Jordan Normal Form.

Assessment Information

Written Exam 50%, Coursework 50%, Practical Exam 0%

Additional Assessment Information

Coursework 50%, Examination 50%

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