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||Core course for Honours Degrees involving Mathematics.
This course showcases the remarkable power of mathematical abstraction. 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. The course also emphasises connections with other parts of mathematics, and will feature applications of the theory to problems, sometimes even beyond mathematics.
In the 'skills' section of this course we continue work with the computer algebra system Maple, learning aspects of it more useful for work in Pure Mathematics. We will learn how to use the rich data structures and programming features of Maple in workshops to investigate in detail some topics in Algebra and students will carry out a group project using Maple and submit Maple work and give a short group presentation on the 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 and Quadratic Forms
1. Basic definitions and examples of inner product spaces.
2. Orthogonal projection; Gram-Schmidt;
3. Quadratic Forms; Sylvester¿s Law of Inertia;
4. 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.
1. Use of a selection of Maple data structures and programming features and using these in different mathematical contexts.
Entry Requirements (not applicable to Visiting Students)
|| Students MUST have passed:
Fundamentals of Pure Mathematics (MATH08064)
||Other requirements|| Students must not have taken :
Information for Visiting Students
Course Delivery Information
|Academic year 2014/15, Available to all students (SV2)
|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 20%, Examination 80%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Honours Algebra (MATH10069) ||3:00|
| 1. Appreciation of abstraction and unity in mathematics, particularly the use of algebra in other topics and the use of other topics in algebra.
2. Ability to work with abstract vector spaces over general fields and linear transformations, described intrinsically or as matrices.
3. Understanding of and facility with fundamental definitions, constructions and theorems in linear algebra, rings and modules, such as kernel and image, quotients, and the first isomorphism theorem.
4. Familiarity with multilinear algebra, mastery of many properties of the determinant, and theoretical and practical knowledge of eigenproblems.
5. Experience of working with inner product spaces, quadratic forms and the spectral theorem.
6. Ability to calculate with the Jordan Canonical Form.
7. Confidence in using some Maple data structures and programming features.
8. The ability to use Maple to investigate suitable topics in pure mathematics.
9. An enhanced understanding of chosen examples obtained by working on them in Maple.
10. Experience of working on a group project.
11. Enhanced presentation skills.
|Graduate Attributes and Skills
|Course organiser||Prof Andrew Ranicki
Tel: (0131 6)50 5073
|Course secretary||Mrs Kathryn Mcphail
Tel: (0131 6)50 4885
© Copyright 2014 The University of Edinburgh - 12 January 2015 4:21 am