Define the function $\ltdefn{R}{\complex{3}}{\complex{4}}$ by \begin{equation*} \lt{R}{\colvector{x_1\\x_2\\x_3}}= \colvector{2x_1-3x_2+4x_3\\x_1+x_2+x_3\\-x_1+5x_2-3x_3\\x_2-4x_3} \end{equation*}
You could verify that $R$ is a linear transformation by applying the definition, but we will instead massage the expression defining a typical output until we recognize the form of a known class of linear transformations. \begin{align*} \lt{R}{\colvector{x_1\\x_2\\x_3}}&= \colvector{2x_1-3x_2+4x_3\\x_1+x_2+x_3\\-x_1+5x_2-3x_3\\x_2-4x_3}\\ &= \colvector{2x_1\\x_1\\-x_1\\0}+ \colvector{-3x_2\\x_2\\5x_2\\x_2}+ \colvector{4x_3\\x_3\\-3x_3\\-4x_3}&&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}\\ &= x_1\colvector{2\\1\\-1\\0}+ x_2\colvector{-3\\1\\5\\1}+\ x_3\colvector{4\\1\\-3\\-4}&&\knowl{./knowls/definition.CVSM.knowl}{\text{Definition CVSM}}\\ &= \begin{bmatrix} 2&-3&4\\ 1&1&1\\ -1&5&-3\\ 0&1&-4 \end{bmatrix} \colvector{x_1\\x_2\\x_3}&&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}} \end{align*}
So if we define the matrix \begin{equation*} B= \begin{bmatrix} 2&-3&4\\ 1&1&1\\ -1&5&-3\\ 0&1&-4 \end{bmatrix} \end{equation*} then $\lt{R}{\vect{x}}=B\vect{x}$. By Theorem MBLT, we can easily recognize $R$ as a linear transformation since it has the form described in the hypothesis of the theorem.