Define $\ltdefn{T}{\complex{3}}{\complex{2}}$ by describing the output of the function for a generic input with the formula \begin{equation*} \lt{T}{\colvector{x_1\\x_2\\x_3}}=\colvector{2x_1+x_3\\-4x_2} \end{equation*} and check the two defining properties. \begin{align*} \lt{T}{\vect{x}+\vect{y}} &=\lt{T}{\colvector{x_1\\x_2\\x_3}+\colvector{y_1\\y_2\\y_3}}\\ &=\lt{T}{\colvector{x_1+y_1\\x_2+y_2\\x_3+y_3}}\\ &=\colvector{2(x_1+y_1)+(x_3+y_3)\\-4(x_2+y_2)}\\ &=\colvector{(2x_1+x_3)+(2y_1+y_3)\\-4x_2+(-4)y_2}\\ &=\colvector{2x_1+x_3\\-4x_2}+\colvector{2y_1+y_3\\-4y_2}\\ &=\lt{T}{\colvector{x_1\\x_2\\x_3}}+\lt{T}{\colvector{y_1\\y_2\\y_3}}\\ &=\lt{T}{\vect{x}}+\lt{T}{\vect{y}} \end{align*} and \begin{align*} \lt{T}{\alpha\vect{x}} &=\lt{T}{\alpha\colvector{x_1\\x_2\\x_3}}\\ &=\lt{T}{\colvector{\alpha x_1\\\alpha x_2\\\alpha x_3}}\\ &=\colvector{2(\alpha x_1)+(\alpha x_3)\\-4(\alpha x_2)}\\ &=\colvector{\alpha(2x_1+x_3)\\\alpha(-4x_2)}\\ &=\alpha\colvector{2x_1+x_3\\-4x_2}\\ &=\alpha\lt{T}{\colvector{x_1\\x_2\\x_3}}\\ &=\alpha\lt{T}{\vect{x}}\\ \end{align*}
So by Definition LT, $T$ is a linear transformation.