## 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 discuss some more consequences of the Steinitz Exchange Lemma, and talk about how to use matrices to recognize a basis of *R*^{n} (or *F*^{n} for any field *F*).

In Lecture 14, we will discuss the* row rank* and *column rank* of a matrix, and show that they are equal. This will conclude our work on Chapter 2, and we will look ahead to Chapter 3, which considers how to describe a linear transformation with respect to different bases.

Slides for Week 7.

Relevant sections of the lecture notes this week are Section 3.1 and Section 3.

Here is (an old version of) Lecture 13.

Here is (an old version of) Lecture 14.

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