Mathematics 2B: Linear Algebra

Short Description
This course covers the fundamentals of linear algebra that are applicable throughout science and engineering, and in particular in the physical, chemical and biological sciences, statistics and other parts of mathematics.

Assessment
One degree examination (80%) (1 hour 30 mins); coursework (20%).
Main Assessment In: December

Course Aims
This course covers the fundamentals of linear algebra that are applicable throughout science and engineering, and in particular in the physical, chemical and biological sciences, statistics and other parts of mathematics. The aim of the first part of the course is to introduce the idea of a finite dimensional vector space, including the concepts of linear independence, basis, dimension and linear map. The relation between linear maps and matrices will be explained, and this will motivate further study of matrices in the second part of the course, in which the determinant, eigenvalues and eigenvectors of a matrix will be studied. Throughout, all new ideas will be illustrated by examples drawn from applications in low dimensions.

Intended Learning Outcomes of Course
Students should be familiar with all definitions and results covered in lectures, should understand the proofs of results, and should be able to apply the results to problems involving the course contents. Students should, moreover, learn to be rigorously logical in their presentation of solutions to problems. By the end of the course, students should be able to (1) handle fluently problems involving matrices and their entries; (2) recognise vector spaces and subspaces over R and C: (3) test sets of vectors for linear independence and spanning properties, and understand methods for obtaining bases for a specified subspace of a vector space; (4) decide whether or not a map between spaces is linear, describe a linear map in matrix form, and calculate various objects (eg image, kernel) associated with a linear map; (5) evaluate determinants recursively and using elementary row and column operations, factorize algebraic determinants, and apply results about determinants in theoretical problems; (6) find the characteristic polynomial of a square matrix, and use it to determine the eigenvalues and eigenvectors of the matrix, and deal with theoretical problems involving eigenvalues.