Page 443, C25
Define the linear transformation and verify that T is a linear transformation:
\[T:\mathbb{C}^{3}\rightarrow\mathbb{C}^{2},\hspace{.5cm}T(\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix})=\begin{bmatrix}2x_1-x_2+5x_3\\-4x_1+2x_2-10x_3\end{bmatrix}\\=x_1\begin{bmatrix}2\\-4\end{bmatrix}+x_2\begin{bmatrix}-1\\2\end{bmatrix}+x_3\begin{bmatrix}5\\-10\end{bmatrix}\\=\begin{bmatrix}2&-1&5\\-4&2&-10\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\]
Any function of this form is a linear transformation.