2.6 Geometry of Matrix Transformations
Matrix transformations, which we explored in the last section, allow us to
describe certain functions. In this section, we will demonstrate how matrix transformations provide a convenient way to describe geometric
operations, such as rotations, reflections, and scalings. We will then explore how matrix transformations are used in computer animation.
Readings
Read Chapter 2 Section 6 in Understanding Algebra by David Austin.
The following video has helpful animations to illustrate the main ideas behind linear transformations:
Linear transformations [latex]T: \mathbb{R}^m \to \mathbb{R}^n[/latex] is a linear transformation if for all vectors [latex]\vec{v}, \vec{w} \in \mathbb{R}^m[/latex] and for all [latex]k \in \mathbb{R}[/latex],
- [latex]T(\vec{v}+\vec{w}) =T(\vec{v}) + T(\vec{w})[/latex]
- [latex]T(k\vec{v}) =kT(\vec{v})[/latex]
Linear Transformations are Matrix Transformations
If [latex]T: \mathbb{R}^m \to \mathbb{R}^n[/latex] is a linear transformation, then it is a matrix transformation. That is, there is an [latex]n\times m[/latex] matrix, [latex]A[/latex], called the standard matrix, so that for all [latex]\vec{v} \in \mathbb{R}^m[/latex], [latex]T(\vec{v}) = A\vec{v}. The standard matrix is given by [latex] A = [T(\vec{e}_1) \dots T(\vec{e}_m)][/latex].