Aims:

To provide a working understanding of matrices and vector spaces for later modules to build on and to teach students practical techniques and algorithms for fundamental matrix operations and solving linear equations.

Content:

Many problems in maths and science are solved by reduction to a system of simultaneous linear equations in a number of variables. Even for problems which cannot be solved in this way, it is often possible to obtain an approximate solution by solving a system of simultaneous linear equations, giving the "best possible linear approximation''.

The branch of maths treating simultaneous linear equations is called linear algebra. The module contains a theoretical algebraic core, whose main idea is that of a vector space and of a linear map from one vector space to another. It discusses the concepts of a basis in a vector space, the dimension of a vector space, the image and kernel of a linear map, the rank and nullity of a linear map, and the representation of a linear map by means of a matrix.

These theoretical ideas have many applications, which will be discussed in the module. These applications include:

• Solutions of simultaneous linear equations
• Properties of vectors
• Properties of matrices, such as rank, row reduction, eigenvalues and eigenvectors
• Properties of determinants and ways of calculating them

Objectives:

Students must understand the ideas of linearly independent vectors, spanning sets and bases of vector spaces. They must also understand the equivalence of linear maps between vector spaces and matrices and be able to row reduce a matrix, compute its rank and solve systems of linear equations. The definition of a determinant in all dimensions will be given in detail, together with applications and techniques for calculating determinants. Students must know the definition of the eigenvalues and eigenvectors of a linear map or matrix, and know how to calculate them.

Books:

The lecture notes will provide comprehensive coverage of the material, but it is all standard foundational mathematics and you can compare how it is covered in many other sources.

David Towers, Guide to Linear Algebra, Macmillan 1988.

Howard Anton, Elementary Linear Algebra, John Wiley and Sons, 1994.

Paul Halmos, Linear Algebra Problem Book, MAA, 1995.

G Strang, Linear Algebra and its Applications, 3rd ed, Harcourt Brace, 1988.