Welcome to Week 8.

In Lecture 15, we will discuss the* kernel* and *image* of a linear transformation, and the analogous concepts of the* rank* and *nullity* of a matrix. We will prove the Rank-Nullity Theorem, which makes an important link between the dimensions of spaces related to a linear transformation.

Here is Lecture 15.

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.