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:
$$ \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a \\ b \end{bmatrix} $$
HintThis translates a point $(x, y)$ by $a$ units horizontally and $b$ units vertically.
5. Rotations
A rotation by angle $\theta$ counterclockwise about the origin is represented by:
$$ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} $$
ExampleTo rotate the point $(1,0)$ by $90^\circ$ counterclockwise:
\[\begin{bmatrix} \cos 90^\circ& -\sin 90^\circ\\\sin 90^\circ&\cos 90^\circ\end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]
Composition of Transformations
Multiple transformations can be combined by multiplying their matrices in the order of application. This is a powerful feature of matrix transformations.
ExampleTo reflect about the y-axis and then rotate by $90^\circ$:
\[\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]
This resulting matrix represents a reflection about the line$y = x$.
NoteThe order of matrix multiplication matters, as matrix multiplication is not commutative.
Iterative Techniques and Fractals
Fractals can be generated using iterative matrix transformations. This involves repeatedly applying a set of transformations to a set of points.
ExampleThe Sierpinski triangle can be generated using three transformations:
\[ T_1 : \begin{bmatrix} 0.5 & 0 \\ 0 & 0.5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \]\[ T_2 : \begin{bmatrix} 0.5 & 0 \\ 0 & 0.5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 0.5 \\ 0 \end{bmatrix} \]\[ T_3 : \begin{bmatrix} 0.5 & 0 \\ 0 & 0.5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 0.25 \\ 0.5 \end{bmatrix} \]
Starting with any point and repeatedly applying these transformations (chosen randomly) results in the Sierpinski triangle fractal.
Determinant and Area
The determinant of a transformation matrix has a geometric interpretation related to area scaling.
For a $2\times2$ matrix $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$, the determinant is:
$$\det A = ad - bc$$
The absolute value of the determinant, $|\det A|$, represents the factor by which the transformation scales areas.
$$\text{Area of image} = |\det A| \times \text{Area of original}$$
ExampleFor a transformation matrix:
\[ A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \]
\[\det A = 2 \times 3 - 0 \times 0 = 6 \]
This transformation scales areas by a factor of 6.
Common MistakeStudents often forget that it's the absolute value of the determinant that matters for area scaling. A negative determinant indicates a change in orientation (e.g., a reflection), but the magnitude still determines the area scaling.
TipWhen studying matrix transformations, always try to visualize the geometric effect of the transformation. This can greatly aid in understanding and problem-solving.
