1. Identify the Move: If the off-diagonals are 0, it is a Scaling. If they are based on Sin/Cos, it is a Rotation.
2. Scaling Rule: $\begin{bmatrix} k_x & 0 \ 0 & k_y \end{bmatrix}$ stretches $x$ by $k_x$ and $y$ by $k_y$.
3. Rotation Rule: $\begin{bmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{bmatrix}$ rotates the vector by angle $\theta$.
4. Magnitude Check: Rotation should never change the length of the vector.
Find the output vector $\mathbf{b} = S\mathbf{x}$ for each scaling matrix.
Uniform Growth: $S = \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 2 \ 5 \end{bmatrix}$
Vertical Stretch: $S = \begin{bmatrix} 1 & 0 \ 0 & 10 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 4 \ 1 \end{bmatrix}$
Contraction (Pruning): $S = \begin{bmatrix} 0.5 & 0 \ 0 & 0.5 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 10 \ 20 \end{bmatrix}$
Use the rotation matrix formula. Remember: $\cos(90^\circ)=0, \sin(90^\circ)=1$.
The 90-Degree Turn: Rotate $\mathbf{x} = \begin{bmatrix} 5 \ 0 \end{bmatrix}$ by 90 degrees counter-clockwise.
The 180-Degree Turn: Rotate $\mathbf{x} = \begin{bmatrix} 3 \ 4 \end{bmatrix}$ by 180 degrees.
Hint: $\cos(180)=-1, \sin(180)=0$.
In the 180-degree turn problem, calculate the magnitude of the input $\sqrt{3^2 + 4^2}$ and the magnitude of your output. Are they the same? Why does rotation preserve "Zeal" (magnitude) but change "Knowledge" (direction)?
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Reflect $\mathbf{x} = \begin{bmatrix} 2 \ 3 \end{bmatrix}$ across the X-axis using $M = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}$.
A soul starts at $\mathbf{x} = \langle 1, 0 \rangle$.
1. First, God rotates them 90 degrees ($R_{90}$).
2. Second, He scales their stature by 5 ($S_5$).
Task: Find the final vector of this soul.
Objective: Explain Rotation and Dilation to a younger student using their own body.
The Activity:
1. Have them stand still. "Dilation: Raise your hands high!"
2. "Rotation: Turn your whole body toward the window!"
The Lesson: "Math can describe how we change. Stretching makes us bigger, and turning makes us see new things."
Response: __________________________________________________________