Matrices

Introduction

A matrix extends the one-dimensional (i.e. single row/column) concept of a vector into two indices (i.e. both rows and columns). Through the definition of a matrix-vector product, matrices act on vectors, transforming them from one basis to another. This is the foundation of linear algebra and forms the building blocks of many more advanced topics, such as eigenvalues and eigenvectors, and the singular value decomposition. In this section, we will first introduce (for many of you, reintroduce) and formalise the concept of a matrix, and how they can be used to represent linear transformations.

We will also discuss some of the basic operations that can be performed on matrices, such as addition, multiplication, and inversion. Finally, we will provide some examples of how matrices are used in practice, along with some exercises to test your understanding of the material. The answers to the exercises are provided, so you can check your work. If the material is familiar to you, there is no need to complete every exercise.