MMath Mathematics and Statistics / Course details

This major core course introduces students to the fundamental concepts of linear algebra, culminating in abstract vector spaces and linear transformations. It covers systems of linear equations, matrices, and some basic concepts of the theory of vector spaces (initially over the real numbers, and then over abstract fields). The subject material is of vital importance in all almost fields of mathematics and in many other areas of science and engineering.

The aims of this course are to introduce the basic ideas and techniques of linear algebra for use in many other lecture courses. The course will also introduce some basic ideas of abstract algebra and techniques of proof which will be useful for future courses in pure mathematics.

On successful completion of this module students will be able to:

• calculate basic matrix and vector operations including sums, products, transposes, powers, Euclidean inner products and norms,
• define and recognise special kinds of matrices including symmetric, skew symmetric, zero, identity, diagonal, upper triangular and elementary matrices,
• apply Gaussian and Gauss-Jordan elimination to solve systems of linear equations, compute (reduced) row echelon forms, test equivalence of matrices, check invertibility of matrices and find inverses,
• define and compute the characteristic polynomial, eigenvalues and eigenvectors of a square matrix;
• define and compute the determinant of a square matrix, using both Gaussian elimination and cofactor expansion,
• define and recognise examples and non-examples of, fields, vector spaces and subspaces of vector spaces,
• apply Gram-Schmidt orthogonalisation to compute an orthogonal or orthonormal basis for a given subspace,
• diagonalize matrices and quadratic forms by finding eigenvalues and appropriate corresponding eigenvectors,
• define linear independence, span and bases, and solve problems involving them,
• define a linear transformation and describe and prove properties of linear transformations,
• define and determine the row space and null space of a matrix and apply the Rank Theorem for matrices,
• solve simple unseen problems, and construct simple proofs of seen and unseen statements, which combine the concepts and methods from this unit with those from the prerequisite unit MATH10101 Foundations of Pure Mathematics A.

1.Linear equations: systems of linear equations, matrices and row echelon form, Gaussian elimination, Gauss-Jordan elimination.

2.Vectors and matrices: linear combinations of vectors, linear independence of vectors, matrix operations, matrix algebra.

3.Elementary matrices: the inverse of a matrix, finding the inverse of a matrix by Gauss-Jordan elimination, LU factorization.

4.Diagonalization of matrices: eigenvalues and eigenvectors, determinants, the characteristic equation, diagonalization of matrices.

5.Orthogonality: orthogonal and orthonormal sets and bases, Gram-Schmidt orthogonalization process, orthogonal matrices, orthogonal complements, Orthogonal Decomposition Theorem, orthogonal projections, applications to matrices, fundamental subspaces, Rank Theorem, orthogonal diagonalization of symmetric matrices, Spectral Theorem, quadratic forms, Principal Axis Theorem, positive definite forms and matrices.

6.Vector spaces: vector spaces and subspaces, definition of vector spaces, examples, subspaces, subspace criterion, sum of subspaces, spanning sets, linear independence, basis, dimension, in a finite-dimensional vector space: every spanning set contains a basis, every linearly independent set can be extended to a basis, any two bases have the same number of elements [Basis Theorem], subspaces are finite dimensional, dimension of the sum of two subspaces, coordinates, change of bases, change-of-basis matrices, Gauss-Jordan method for computing change-of-basis matrices.

7.Linear transformations: definition and examples, composition, inverse, kernel and range of a linear transformation, kernel, range, rank and nullity, Rank Theorem, one-to-one and onto linear transformations, universal property of vector spaces, isomorphisms of vector spaces, matrices of linear transformations, definition and elementary properties, matrices of composites and inverses, change-of-basis and similarity, diagonalization of linear transformations.

Attendance and submission of work at supervisions: weighting 10%

Examination: weighting 90%

Weekly marked coursework will provide an opportunity for students to receive feedback on their written work. Feedback supervisions provide an opportunity for further feedback on understanding. Students can also seek feedback directly from the lecturers, during weekly office hours.

All the textbooks below are useful for further examples and clarification as well as applications.

Linear Algebra: a Modern Introduction, D. Poole, published by Thomson, 3rd edition 2010, Linear Algebra, J. Fraleigh, published by Addison-Wesley, 3rd edition 1995

Elementary Linear Algebra, H. Anton, published by Wiley, 5th edition 1987

Linear Algebra and its Applications, D. Lay, published by Addison-Wesley, 4th edition 2012

3000 Solved Problems in Linear Algebra, S. Lipschutz, published by McGraw-Hill, 1988

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  Please see Blackboard / course unit related emails for any further updates