We are tasked with verifying whether \[ T:\mathbb{C}^3 \rightarrow \mathbb{C}^2 \] such that \[ T \cdot \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} = \begin{bmatrix} 2x_1 -x_2 +5x_3 \\ -4x + 2x_2-10x_3 \end{bmatrix} \]
Since we can rearrange the problem as the sum of linearly independent vectors:
\[ = x_1 \cdot \begin{bmatrix} 2 \\ -4 \end{bmatrix} + x_2 \cdot \begin{bmatrix} -1 \\ 2 \end{bmatrix} + x_3 \cdot \begin{bmatrix} 5 \\ -10 \end{bmatrix} \]
We know that this function is a linear transformation by Theorem MBLT.