Sections 1.8 and 1.9 in the textbook are closely related, and it makes sense to discuss them together. We define the idea of a matrix transformation and discuss some standard low-dimension matrix dimensions from a geometric point of view. Then we show that every matrix transformation is in fact a linear transformation, and use that to discuss linear transformations in general. We come full circle by showing that also every linear transformation is in fact a matrix transformation and we show how to find the associated matrix.

Outline:

Definition of matrix transformation
Some low-dimensional geometry of certain matrix transformations
Definition of linear transformation
Every matrix transformation is a linear transformation, and every linear transformation is a matrix transformation
Computing the matrix associated to a linear transformation

Video lecture:

Section 1.8
Section 1.9 Part 1
Section 1.9 Part 2

Notes from the lecture:

Section 1.8 Introduction to Linear Transformations (Summer 2020).pdf
Section 1.9 The Matrix of a Linear Transformation Part 1 (Summer 2020).pdf
Section 1.9 The Matrix of a Linear Transformation Part 2 (Summer 2020).pdf