Content

Linear algebra is a branch of mathematics aimed at studying vector spaces and linear transformations between them. In the first part of this module we will take a closer look at linear transformations and discuss how they are related to matrices. We will then introduce the determinant of linear maps and see how it detects their invertibility. Many problems in linear algebra lead to systems of linear equations and we will see how to solve them using elementary row operations. Another important chapter in this course deals with the diagonalisation of linear endomorphisms and square matrices. We will not only introduce eigenvectors, eigenvalues and eigenspaces, but also see applications of those concepts in computer science and mathematics. Not all endomorphisms are diagonalisable. It is however always possible to find a basis, such that the corresponding matrix is "almost diagonal". This Jordan normal form of linear transformations will be defined, and we will discuss how to compute it in examples. At the end of the course we will look at generalisations of the standard inner products on $\mathbb{R}^n$ and $\mathbb{C}^n$, which will motivate the study of bilinear and hermitian forms on real, respectively complex, vector spaces.

Lecture Notes

The lecture notes can be downloaded by following the link below. Note that these notes are from the version of the course given in the autumn semester 2019.

Exercises

The following are the problem sheets and the mock exam that I created for this course.