Undergraduate Course: Honours Algebra (MATH10069)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 3 Undergraduate)
||Availability||Available to all students
|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.
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.
1. 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 Modules
1. 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 Eigenvalues
1. 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 Spaces
1. 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 Form
1. The Jordan Normal Form.
2. Applications of the Jordan Normal Form.
Information for Visiting Students
|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.
|High Demand Course?
Course Delivery Information
|Academic year 2022/23, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 35,
Seminar/Tutorial Hours 10,
Supervised Practical/Workshop/Studio Hours 10,
Summative Assessment Hours 3,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 50%, Examination 50%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||MATH10069 Honours Algebra||2:00|
On completion of this course, the student will be able to:
- Work/compute with the objects introduced in the class (abstract vector spaces, rings, modules), the notions associated to the maps (linear transformations/homomorphism) between them (kernel, images, determinants), constructions with them (quotients), forms on vector spaces (inner products) and to make explicit calculations (matrices).
- Demonstrate an understanding of the topics covered in the course, especially the first isomorphism theorem, properties of the determinant, properties of eigenvalues, the spectral theorem, by explaining the key aspects of their proofs and by proving variations or extensions of them.
- Apply these results to examples that have not been covered in course.
- Write and test short procedures to carry out calculations in abstract algebra using a computer algebra system.
- Work collaboratively to investigate an application of abstract algebra using a computer algebra system, and to communicate the findings succinctly,
|Graduate Attributes and Skills
|Course organiser||Dr Johan Martens
Tel: (0131 6)51 7759
|Course secretary||Miss Greta Mazelyte