Welcome to Week 8.

Lecture 15 will not be run in person, but via a pre-recorded video, which is below. The main theme here is how to move between different bases, when describing an element of a finite-dimensional vector space via its coordinates. If we have two different bases for a vector space* V* of dimension *n*, then an element *v* of *V* has coordinates with respect to each basis. As usual in linear algebra, there is a matrix calculation that takes us from one of these expressions to the other. What is needed are the coefficients in the expressions for one set of basis elements as linear combinations of the other.

Lecture 15 also discusses how to write linear transformations as matrices in general, with respect to given bases.

Here is Lecture 15.

In Lecture 16 (in person on Friday), we will discuss the concepts of the kernel and image of a linear transformation, and the very useful *Rank-Nullity Theorem.* We will also start to talk about the relation of *similarity* for square matrices, which will continue to be a theme in Week 9.

Slides for Week 8.

Old recordings of lectures on this content are below.

Slides for Lecture 15 (and part of Lecture 16).

In Section 3.3 we will discuss linear transformations from a space to itself, which correspond to square matrices. We will discuss how matrices that describe the same transformation with respect to different bases are related to each other algebraically. This will bring us to the concepts of *similarity* and *diagonalizability* of matrices, and on to eigenvectors, which will be next week’s theme.

Here is Lecture 16.

Slides for Lecture 16.