Please note: This course will be taught online only. In person study is not available for this course. 

Jessica Claridge is a lecturer in mathematics in the School of Statistics, Mathematics and Actuarial Sciences at the University of Essex. She has taught at the Essex Summer School since 2017. She received a PhD in Mathematics on the area of Network Coding from Royal Holloway, University of London in 2017. Her research interests include information theory, coding theory and mathematics education.

Course Content
The course will introduce participants to the essential concepts of linear algebra required for modern data analysis, with the goal of building confidence in the mathematical tools that underpin statistical modelling, machine learning, and quantitative social science. This course is aimed at being accessible to those who have not studied linear algebra before or have limited experience in this area and want to improve their understanding and confidence. Specifically, the course will explore the following topics:

  • Vectors and matrices
  • Gaussian elimination and systems of linear equations
  • Linear least squares
  • Multiple regression problems
  • Applications to statistics and data analysis

 
Course Objectives
Upon completion of this course, participants will have conceptual and practical understanding of vectors and matrices and their use within data analysis methods such as linear least squares and multiple regression.

Course Prerequisites
This is an introductory and highly accessible course. No specific mathematical background is required.

Representative Background Reading
Since this is an introductory course, participants are not required to complete any prior reading.

Required texts
All relevant materials will be provided within the course lecture notes.

Introduction to Linear Algebra
Across five sessions, participants will develop an intuitive understanding of vectors, matrices, systems of linear equations, and the powerful least-squares techniques that form the basis of regression analysis. By the end of the course, learners will be equipped with the mathematical tools needed to engage more deeply with data driven research and analytical methods.

Day 1 – Vectors, Matrices, and operations
We introduce scalars, vectors, and matrices, and explain how these objects represent data and transformations. We will study operations such as vector addition, dot products, and matrix multiplication, and explore how matrices are used to structure and manipulate data.

Day 2 – Solving Linear Systems with Gaussian Elimination
Learn to express systems of linear equations in matrix form and use Gaussian elimination and row-reduction techniques to find solutions. We examine how to interpret different solutions types (unique solutions, no solution, many solutions) and how these may arise in real data.

Day 3 – Linear Independence, Rank, and Least Squares
Explore the ideas of linear combinations, independence, and the rank of a matrix, and understand how this relates to the existence of solutions to systems of linear equations. We introduces the linear least squares method for solving overdetermined systems which is the foundation for regression analysis. We consider the linear least square solution to be the ‘best-fit’ solution when exact solutions do not exist.

Day 4 – Multiple Regression
Explore how matrices are used to express multiple regression models, learn to solve them efficiently, and understand the meaning of projections and residuals. This session highlights the power of linear algebra as a foundation for rigorous data analysis.

Day 5 – Further Applications to Data Analysis
The final session demonstrates further applications of linear algebra, such as dimension reduction, correlation analysis, and regularisation. Participants then apply the techniques seen to problems, reinforcing understanding through hands-on analysis.