The aim of this module is to introduce students to the main ideas of Linear Algebra and its many applications. The emphasis is on developing the student's ability to perform calculations on and with matrices, particularly 2x2 and 3x3 matrices, and on and with vectors in 2 and 3 dimensions. These ideas are then extended to higher dimensions.

Matrices: introduction to matrices, matrix algebra, transpose of a matrix, symmetric matrices, invertible matrices and their inverses, determinants. Vectors in 2 and 3 dimensions: geometric interpretation of vectors, vector arithmetic, Euclidean norm, Euclidean scalar product, angle, orthogonality, projections, cross product and its uses in the study of lines and planes in 3 dimensions. Lines and planes in 3-dimensional space: parametric equation of a line, distance between a point and a line, point-normal form and general form of the equation of a plane, distance between a point and a plane. Extension to vectors in n dimensions; Systems of linear equations and their solution: Gaussian elimination methods (Gauss, Gauss-Jordan) and inverse matrix method; Matrices acting on vectors: eigenvalues and eigenvectors particularly in 2 and 3 dimensions. Applications: least squares fit, rotation matrices.

The student will be able to carry out matrix algebra operations on matrices of any order including the evaluation of determinants of square matrices and the calculation of eigenvalues and eigenvectors. The student will be able to apply Gaussian elimination to a system of linear equations in order to determine its solution and use Gaussian elimination to find the inverse of an invertible matrix. The student will also be able to apply the inverse matrix method to solve linear systems, where appropriate. The student will be able to carry out vector space operations on vectors of 2 and 3 dimensions including inner product, cross product and norm calculations. The student will be able to use vectors to solve problems in 3-dimensional geometry involving lines and planes. The student will learn how the concepts learnt can be generalised to n-dimensional spaces. The student will be able to use matrices in a variety of applications.

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Normal weekly lectures and tutorials. Mid-term exam(s) and final exam. Graduates, on successful completion of the module, will be well grounded in the foundations of linear algebra and will show confidence in their choice of appropriate method for solving problems and proving basic results in linear algebra. Graduates will be expected to be proactive and collaborative in tackling problems proposed in the tutorials.