This is a simple example of the Coordinatization Principle, depending only on the fact that coordinatizing is an invertible linear transformation (Theorem VRILT). Suppose we have a linear combination to perform in $M_{32}$, the vector space of $3\times 2$ matrices, but we are adverse to doing the operations of $M_{32}$ (Definition MA, Definition MSM). More specifically, suppose we are faced with the computation \begin{equation*} 6 \begin{bmatrix} 3 & 7\\ -2 & 4\\ 0 & -3 \end{bmatrix} +2 \begin{bmatrix} -1 & 3\\ 4 & 8\\ -2 & 5 \end{bmatrix} \end{equation*}
We choose a nice basis for $M_{32}$ (or a nasty basis if we are so inclined), \begin{equation*} B=\set{ \begin{bmatrix}1&0\\0&0\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\1&0\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\0&0\\1&0\end{bmatrix},\, \begin{bmatrix}0&1\\0&0\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\0&1\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\0&0\\0&1\end{bmatrix} } \end{equation*} and apply $\vectrepname{B}$ to each vector in the linear combination. This gives us a new computation, now in the vector space $\complex{6}$, which we can compute with operations in $\complex{6}$ (Definition CVA, Definition CVSM), \begin{align*} 6\colvector{3\\-2\\0\\7\\4\\-3}+2\colvector{-1\\4\\-2\\3\\8\\5}=\colvector{16\\-4\\-4\\48\\40\\-8} \end{align*}
We are after the result of a computation in $M_{32}$, so we now can apply $\ltinverse{\vectrepname{B}}$ to obtain a $3\times 2$ matrix, \begin{align*} &16\begin{bmatrix}1&0\\0&0\\0&0\end{bmatrix}+ (-4)\begin{bmatrix}0&0\\1&0\\0&0\end{bmatrix}+ (-4)\begin{bmatrix}0&0\\0&0\\1&0\end{bmatrix}+ 48\begin{bmatrix}0&1\\0&0\\0&0\end{bmatrix}+ 40\begin{bmatrix}0&0\\0&1\\0&0\end{bmatrix}+ (-8)\begin{bmatrix}0&0\\0&0\\0&1\end{bmatrix}\\ &\quad\quad=\ \begin{bmatrix}16&48\\-4&40\\-4&-8\end{bmatrix} \end{align*} which is exactly the matrix we would have computed had we just performed the matrix operations in the first place. So this was not meant to be an easier way to compute a linear combination of two matrices, just a different way.