Suppose that $\ltdefn{T}{\complex{4}}{\complex{3}}$ is defined by \begin{equation*} \lt{T}{\colvector{x_1\\x_2\\x_3\\x_4}}= \colvector{x_1+2x_2-x_3+2x_4\\x_1+5x_2-3x_3+x_4\\-2x_1+3x_2-4x_3+2x_4} \end{equation*}
For the sake of an example, choose $\alpha=2$, so by Definition LTSM, we have \begin{align*} \lt{\alpha T}{\colvector{x_1\\x_2\\x_3\\x_4}} &=2\lt{T}{\colvector{x_1\\x_2\\x_3\\x_4}} =2\colvector{x_1+2x_2-x_3+2x_4\\x_1+5x_2-3x_3+x_4\\-2x_1+3x_2-4x_3+2x_4}\\ &=\colvector{2x_1+4x_2-2x_3+4x_4\\2x_1+10x_2-6x_3+2x_4\\-4x_1+6x_2-8x_3+4x_4} \end{align*} and by Theorem MLTLT we know $2T$ is also a linear transformation from $\complex{4}$ to $\complex{3}$.