## Moving between different bases

Welcome to Week 7.

This week we will start on Chapter 3 of the lecture notes, on linear transformations, eigenvectors and similarity. We will be continuing on the theme of bases, and a theme of this chapter will be to explore how some bases are much better than others for describing a particular linear transformation – either to get some insight into its geometric behaviour or just to do calculations. As usual, almost everything can be translated into the algebra of matrices.

In Lecture 13, we will use an example from R^{3} to describe how to write coordinates of a vector with respect to different bases. There is a consistent method for this and it boils down to matrix-vector multiplication.

Here is Lecture 13.

Slides for Lecture 13, without annotation.

In Lecture 14, we will discuss the matrix of a linear transformation from a vector space U to a vector space W, with respect to a choice of bases for U and W. Lecture 14 will also introduce the *Rank-Nullity Theorem*, probably we will come back to that in Week 8.

Here is Lecture 14.

Slides for Lecture 14, without annotation

Relevant sections of the lecture notes this week are Section 3.1 and Section 3.2.