Overview

Linear algebra and optimization form the working language of many models in statistics, economics, finance and machine learning. This course focuses on the structures that recur across those fields rather than on abstract theorem proving for its own sake.

Learners study vectors, matrix transformations, rank, projection, eigenstructure, gradients and constrained optimization through modelling problems. The course repeatedly asks what a mathematical operation means for a real estimation or decision problem.

Particular attention is given to high-dimensional settings, where geometric intuition based on two or three dimensions can become misleading.

Learning objectives

By the end of the course, learners should be able to:

• work confidently with vectors, matrices and linear systems;
• interpret projection, rank and eigenstructure in modelling problems;
• formulate unconstrained and constrained optimization problems;
• use gradients and first-order conditions to characterize solutions;
• explain why high-dimensional problems behave differently from low-dimensional intuition;
• connect linear algebra and optimization to regression and machine learning.

Intended participants and prerequisites

Course structure

The course is organised into six units. Each unit combines concise explanation, worked examples and short analytical exercises.

Unit	Subject	Principal questions
1	Vectors and geometry	How do vectors represent observations, parameters and directions?
2	Matrices and linear systems	How do transformations, rank and inversion organize multivariate relationships?
3	Projection and least squares	Why is regression fundamentally a projection problem?
4	Eigenvalues and structure	What do eigenvectors reveal about dynamics, variance and repeated transformations?
5	Optimization	How do gradients, curvature and constraints determine candidate solutions?
6	High-dimensional reasoning	Why do intuition, regularization and computation change as dimension grows?

Teaching and assessment

Teaching is delivered through concise lectures, technical notes, worked examples and guided exercises. AI-supported interaction may be used to explain concepts, generate additional practice and challenge a learner's reasoning, but assessment remains based on the learner's demonstrated understanding.

Component	Purpose	Weight
Short exercises	Check understanding of the principal concepts	40%
Applied investigation	Use the course framework on a new problem or dataset	30%
Final assessment	Integrated reasoning across the course	30%

A certificate of completion is issued only when the required assessments have been completed at the stated standard. Participation alone does not constitute completion.

Faculty

Materials and technical requirements

Learning environment

Learners need a current web browser and access to the Gordon learning platform. Where a course uses code, spreadsheets, datasets or external readings, the relevant requirements are identified inside the unit.

Course materials

Required readings, datasets and practice materials are identified within each unit. Public courses are designed so that required core materials can be accessed without a separate tuition charge.