Problem
Determine a linear transformation function that would rotate the pink figure by $105^{\circ}$ clockwise and would dilate it by a factor of 0.75 . The general form of the linear transformation should be $L(x, y)=(a x+b y, c x+d y)$. You must round all coefficients to four decimal places. The resulting figure should be the green shape.
Solution
Step 1 ：Convert 105 degrees to radians: \(\theta_{rad} = 1.8326\)
Step 2 ：Find the rotation matrix: \(\begin{bmatrix} -0.2588 & 0.9659 \\ -0.9659 & -0.2588 \end{bmatrix}\)
Step 3 ：Apply the dilation factor of 0.75: \(\begin{bmatrix} -0.1941 & 0.7244 \\ -0.7244 & -0.1941 \end{bmatrix}\)
Step 4 ：\boxed{L(x, y) = (-0.1941x + 0.7244y, -0.7244x - 0.1941y)}
