Example STLT Sum of two linear transformations

Suppose that $\ltdefn{T}{\complex{2}}{\complex{3}}$ and $\ltdefn{S}{\complex{2}}{\complex{3}}$ are defined by \begin{align*} \lt{T}{\colvector{x_1\\x_2}}=\colvector{x_1+2x_2\\3x_1-4x_2\\5x_1+2x_2} && \lt{S}{\colvector{x_1\\x_2}}=\colvector{4x_1-x_2\\x_1+3x_2\\-7x_1+5x_2} \end{align*}

Then by Definition LTA, we have \begin{align*} \lt{(T+S)}{\colvector{x_1\\x_2}}&=\lt{T}{\colvector{x_1\\x_2}}+\lt{S}{\colvector{x_1\\x_2}}\\ &=\colvector{x_1+2x_2\\3x_1-4x_2\\5x_1+2x_2}+\colvector{4x_1-x_2\\x_1+3x_2\\-7x_1+5x_2} =\colvector{5x_1+x_2\\4x_1-x_2\\-2x_1+7x_2} \end{align*} and by Theorem SLTLT we know $T+S$ is also a linear transformation from $\complex{2}$ to $\complex{3}$.