Question 20
(a) Write down the matrix of the linear transformation \( f \) that maps \( (1,0) \) to \( (2,2) \) and \( (0,1) \) to \( (-1,3) \).
(b) Show that \( f \) is invertible and find the matrix of \( f^{-1} \).
(c) A triangle \( T \) has vertices at \( (2,0),(0,2) \) and \( (0,0) \). Find the area of \( f(T) \).
(d) Let \( g \) be the affine transformation given by
\[
g(\mathbf{x})=\left(\begin{array}{rr}
1 & 0 \\
3 & -2
\end{array}\right) \mathbf{x}+\binom{1}{1}
\]
Find the rule of \( f^{-1} \circ g \) in the form
\[
\left(f^{-1} \circ g\right)(\mathbf{x})=\mathbf{A x}+\mathbf{a}
\]
where \( \mathbf{A} \) is a \( 2 \times 2 \) matrix and a is a \( 2 \times 1 \) column vector.
Simplify your answer as much as possible.
