Use basic linear transformations: dilations of the form (𝑥, 𝑦) → (𝑎𝑥, 𝑏𝑦), rotations about the origin and reflection in a line that passes through the origin, and the representations of these transformations by 2 × 2 matrices. dilation of factor 𝑎 parallel to the 𝑥-axis and factor 𝑏 parallel to the 𝑦-axis: [𝑎 0 0 𝑏] rotation of angle 𝜃 anticlockwise about the origin: [cos(𝜃) − sin(𝜃) sin(𝜃) cos(𝜃)] reflection in the line 𝑦 = 𝑥 tan(𝜃): [cos(2𝜃) sin(2𝜃) sin(2𝜃) − cos(2𝜃)]

A linear transformation \(T\) is applied to the point \(P(3, 2)\), where \(T\) is a dilation of factor 4 parallel to the \(x\)-axis and factor 2 parallel to the \(y\)-axis, followed by a rotation of \(60^\circ\) anticlockwise about the origin. a) Determine the \(2 \times 2\) matrix that represents the dilation transformation. b) Determine the \(2 \times 2\) matrix that represents the combined transformation \(T\). c) Use your result from part b) to determine the coordinates of the image point \(P'\) after the transformation \(T\) is applied to \(P\).

A triangle with vertices \(A(2, 1)\), \(B(4, 1)\) and \(C(3, 3)\) undergoes three successive transformations as shown in the table below. Determine the coordinates of the image of vertex \(C\) after all three transformations have been applied.

Use the transformation matrix \(\begin{bmatrix} 3 & 0 \\ 0 & -2 \end{bmatrix}\) to determine the image of the point \((4, 5)\).