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.

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 $T : R^{m} \to R^{n}$ is a linear transformation if for all vectors $\vec{v}, \vec{w} \in R^{m}$ and for all $k \in R$ ,

Linear Transformations are Matrix Transformations

If

T : R^{m} \to R^{n}

is a linear transformation, then it is a matrix transformation. That is, there is an

n \times m

matrix,

A

, called the standard matrix, so that for all

\vec{v} \in R^{m}

,

T (\vec{v}) = A \vec{v} . T h e s t a n d a r d m a t r i x i s g i v e n b y [l a t e x] A = [T ({\vec{e}}_{1}) \dots T ({\vec{e}}_{m})]

.

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