BSc Mathematics / Course details

Linear algebra is one of the most applicable areas of mathematics, both within mathematics itself as well as to other areas of knowledge, from Engineering to Economics.  This course provides a solid introduction to this area and introduces many properties of matrices, such as determinants and eigenvalues, as well as an introduction to abstract vector spaces. Almost all the results in the course are given formal proofs giving a precise understanding of their validity

The unit aims to introduce the basic ideas and techniques of linear algebra for use in many other lecture courses in the undergraduate mathematics programmes and beyond. It also aims to illustrate the ubiquitous nature of the subject throughout the sciences. The course will also introduce some basic ideas of abstract algebra and techniques of proof which will be useful for future courses in mathematics, and in addition it will introduce appropriate mathematical software to enable students to perform calculations on matrices that are inaccessible by hand.

On the successful completion of the course, students will be able to:

• Manipulate matrices and vectors though their basic operations, including linear combination, multiplication, transpose and inversion; define, recognize and be able to use special classes of matrices, including symmetric, skew-symmetric, triangular, orthogonal and unitary matrices;
• solve systems of linear equations using Gaussian and Gauss-Jordan elimination and LU decomposition;
• compute determinants, eigenvalues and eigenvectors of square matrices and know the relation of eigenvalues with determinant and trace; diagonalize matrices by finding eigenvalues and eigenvectors;
• define linear independence, span and basis, and solve problems involving their properties, including use of the Gram-Schmidt procedure and the QR decomposition of a matrix
• construct proofs of properties of fields, vector spaces and linear maps, recognize subspaces and calculate their dimension;
• define, recognize and construct proofs of simple properties of linear transformations; calculate the matrix for a linear transformation with respect to a given basis; calculate the kernel and image of a linear transformation;
• define and determine the row space, column space and rank of a matrix, and apply the Rank Theorem;
• define inner products, norms and isometries on vector spaces, and construct proofs of simple properties of these;
• solve simple unseen problems, and construct simple proofs of seen and unseen statements, which combine the concepts and methods from this unit.

There is a supervision;each week which provides an opportunity for students; work to be marked and discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback; Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

G. Strang, Introduction to Linear Algebra (5th Ed). Wellesley-Cambridge Press, 2016.

J. Hefferon, Linear Algebra. Available (freely) online from https://hefferon.net/linearalgebra/

Linear Algebra, by E.S.Meckes and M.W.Meckes, published by Cambridge University Press