Linear Operators
Chapter 4 • Section 4-4
"Spin it, flip it, stretch it... matrices in action! 🔄"
🔄 Rotations in $\mathbb{R}^2$
Rotation Matrix $R_\theta$
To rotate a vector counterclockwise by angle $\theta$:
$$ R_\theta = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}
$$
🪞 Reflections
X-Axis Reflection
$$ Q_x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $$
Y-Axis Reflection
$$ Q_y = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} $$
Reflection across line $y=mx$
$$ Q_m = \frac{1}{1+m^2} \begin{bmatrix} 1-m^2 & 2m \\ 2m & m^2-1 \end{bmatrix} $$
🔗 Multiple Actions (Composition)
If we apply transformation $A$ then transformation $B$, the combined transformation is $BA$.
Order matters! The matrix on the right acts first.
$$ \text{Action}(\vec{x}) = B(A\vec{x}) = (BA)\vec{x} $$
Two Rotations
Rotating by $\alpha$ then $\beta$ is the same as rotating by $\alpha + \beta$.
$$ R_\beta R_\alpha = R_{\alpha+\beta} $$
Two Reflections
Reflecting across line $L_1$ then $L_2$ (angle $\theta$ between them) is a rotation by $2\theta$.
$$ Q_2 Q_1 = R_{2\theta} $$
Transformation Playground
Apply transformations to the "F" shape.
Current Matrix: Identity
🤔 Logic Check Win $20
What is the result of reflecting across the X-axis followed by reflecting across the Y-axis?