Jessica Claridge graduated from Queen Mary, University of London with an MSci in Mathematics in 2012. She then went on to complete a PhD in Mathematics at Royal Holloway, University of London in 2017. She joined the University of Essex in 2016 as a Tutor and part-time Lecturer in Mathematics. As part of her role she is the Area Coordinator for the Further Mathematics Support Programme. Her research interests include Network Coding, Combinatorics and Finite Fields.

Course Content This component of the course focuses on solving systems of linear equations by Gaussian elimination; inverse matrices and singularity; vector spaces and subspaces; linear dependence, dimension, and rank; matrix eigenvalues and eigenvectors. It also considers the application of these topics to the linear regression problem. Finally, the course considers the method of linear algebra called ’Singular Value Decomposition’ (SVD) which lies at the heart of many useful applications.

Course Objectives To provide participants with the essentials of linear algebra required for the study of multivariate analysis. Emphasis is placed on the relationship between the algebra and geometry.

Course Prerequisites For participants who have not taken the second half of Mathematics for Social Scientists, Part 2, some familiarity with matrix arithmetic is helpful. However, there is a short summary of basic matrix arithmetic at the end of the part 3 course notes which is adequate for this component of the course and which will be briefly reviewed.

Recommended Reading For a review of the concepts listed in the prerequisites we recommend the Matrices and Vectors quick reference leaflets which can be found by following the leaflets link from http://www.mathscentre.ac.uk/students.php. Note that the site also has learning resources available for these and other basic mathematical topics.

Additional Reading Linear Algebra a Modern Introduction, 4th Edition by David Poole, published by Cengage 2015. ISBN 9781285463247

Students entering the course should be familiar with the basic ideas of arithmetic and algebra (addition, subtraction, multiplication, division, use of brackets, positive negative and fractional powers, and the solution of simple equations) as described, for example, in Haeussler, E.F., Paul, R.S. and Wood,R., Mathematical Analysis for Business, Economics and the Life and Social Sciences, 11th ed., Pearson (pp. 2-19, 30-34 and 147-151). Students should possess a basic scientific calculator.

Students may take any or all of the three Parts of the course, but anyone taking Parts 2 or 3 will be presumed to be familiar with the material covered in the preceding Parts.

Lecture 1:Operations on vectors.

Lecture 2: Linear Independence and vector spaces.

Lecture 3: Geometric Interpretation of Matrices.

Lecture 4:Gaussian Elimination and Systems of Linear Equations.

Lecture 5:Matrix Inverse and Solutions of Linear Equations

Lecture 6:Linear Least Squares.

Lectures 7:Multiple Regression Problems.

Lectures 8:Eigenvalues, Eigenvectors and Symmetric Matrices.

Lectures 9: General Singular Value Decomposition (SVD) and applications

Lectures 10: Overview and Plenary Session

Reading list: The book by Haeussler, Paul and Wood referred to above is an acceptable text for Sessions 1 and 2 of the course. Students may also find the following books useful, although it is not necessary to read them before the course:

+ *Arya,J.C. and Lardner,R.W., Mathematical Analysis for Business, Economics and the Life and Social Sciences, 4th ed., Pearson + *Booth, D.J. Foundation Mathematics. Addison Wesley + *Dowling, E.T. Introduction to Mathematical Economics. Schaum’s Outline Series. McGraw Hill. (This book contains many worked examples.) + *Black, J. and Bradley, J.F. Essential Mathematics for Economists. Wiley. +Chiang, A.C. Fundamental Methods of Mathematical Economics. McGraw Hill. *Croft, A. and Davison, R. Foundation Maths. Longman. *Green, P.E. Mathematical Tools for Applied Multivariate Analysis. Academic Press. + *Grossman, S.I. Elementary Linear Algebra. Wadsworth. *Mizrahi, A. and Sullivan, M. Mathematics, an Applied Approach. Wiley. *Nicholson, R.H. Mathematics for Business and Economics. McGraw Hill. *Page, S., Berry, J. and Hampson, H. Mathematics, a Second Start. Prentice Hall. *Smith, K.J. College Mathematics and Calculus. Brooks/Cole.

*indicates that this book is in the Summer School Library +indicates that this book is in the University Library

The list contains good books for people wanting to read around the topics we will be covering during the course but they are not required reading and will not be referred to directly in the lectures so purchasing them is entirely optional.

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