Matrix Transformations in Two Dimensions
Matrix transformations are a powerful tool in linear algebra that allow us to represent and perform geometric transformations on points in a coordinate system.
Basic Transformation Matrix
The general form of a 2D matrix transformation is:
$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} e \\ f \end{bmatrix} $$
HintHere, $\begin{bmatrix}x \\ y\end{bmatrix}$ represents the original point, and the resulting coordinates after transformation are obtained by multiplying this vector with the $2\times2$ matrix and adding the translation vector $\begin{bmatrix}e \\ f\end{bmatrix}$.
NoteThe $2\times2$ matrix represents linear transformations such as rotation, reflection, and scaling, while the added vector represents translation.
Types of Transformations
1. Reflections
Reflections can be represented using matrices. For example, reflection about the y-axis:
$$ \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} $$
ExampleThe point $(3,2)$ reflected about the y-axis becomes $(-3,2)$:
\[\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} -3 \\ 2 \end{bmatrix} \]
2. Stretches
Horizontal and vertical stretches can be represented by scaling the corresponding coordinate:
- Horizontal stretch by factor k: $\begin{bmatrix}k & 0 \\ 0 & 1\end{bmatrix}$
- Vertical stretch by factor k: $\begin{bmatrix}1 & 0 \\ 0 & k\end{bmatrix}$
3. Enlargements
An enlargement is a uniform scaling in all directions:
$$ \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} $$
where $k$ is the scale factor.
4. Translations
Translations are represented by the addition of a vector:
