Answer
See below
Work Step by Step
a) Obtain $T(e^{2x})=2e^{2x}\\T(e^{-3x})=-3e^{-3x}$
then we have:
$T(e^{2x})=2e^{2x}=2.e^{2x}+0.e^{-3x}\\
T(e^{-3x})=-3e^{-3x}=0.e^{2x}-3.e^{-3x}$
as $[T(e^{2x})]_C=\begin{bmatrix}
2\\
0
\end{bmatrix}\\
[T(e^{-3x})]_C=\begin{bmatrix}
0 \\-3
\end{bmatrix}$
Hence, $[T]_B^C=\begin{bmatrix}
2 & 0 \\
0 & -3\end{bmatrix}$
b) Obtain $T(e^{2x}-3e^{-3x})=2e^{2x}+9e^{-3x}\\T(2e^{-3x})=-6e^{-3x}$
then we have:
$T(e^{2x}-3e^{-3x}))=2e^{2x}=9.(e^{2x}+e^{-3x})+7.(-e^{2x})\\
T(e^{-3x})=-6e^{-3x}=-6(e^{2x}+e^{-3x})-6.e^{2x}$
as $[T(e^{2x})]_C=\begin{bmatrix}
9\\
7
\end{bmatrix}\\
[T(e^{-3x})]_C=\begin{bmatrix}
-6 \\-6
\end{bmatrix}$
Hence, $[T]_B^C=\begin{bmatrix}
9 & -6 \\
7 & -6\end{bmatrix}$
