Define the linear transformation \(T:C^{3}\longrightarrow C^{2}\),
\[ T= \bigg(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\bigg)\quad = \begin{bmatrix} 2x_{1} & +x_{2} & +5x_{3} \\ -4x_{1} & +2x_{2} & -10x_{3} \\ \end{bmatrix} \]
In our first step we separate the matrix into 3 subvectors (i.e.\(x_{1},x_{2}...\)) based on the contents of \(T\) \[ = \begin{bmatrix} 2x_{1} \\ -4x_{1} \\ \end{bmatrix}\quad + \begin{bmatrix} -x_{2} \\ 2x_{2} \\ \end{bmatrix}\quad + \begin{bmatrix} -5x_{3} \\ -10x_{3} \\ \end{bmatrix}\quad \]
Secondly, we extract the contents of \(T\) and place them outside of the subvectors…
\[ =x_{1} \begin{bmatrix} 2 \\ 4 \\ \end{bmatrix}\quad +x_{2} \begin{bmatrix} -1 \\ 2 \\ \end{bmatrix}\quad +x_{3} \begin{bmatrix} 5 \\ -10 \\ \end{bmatrix}\quad \]
Lastly, we arrive at the following matrix
\[ =x_{1} \begin{bmatrix} 2 & -1 & 5\\ 4 & 2 & -10\\ \end{bmatrix}\quad \]