I think you're missing the first step, which is, formalizing a problem using matrices and vectors in the first place.

Correct me if i'm wrong, but i can easily imagine that people have been solving those types of issues manualy for centuries before finding the "shortcut" or representing transforms using matrices and figuring the rules of matrix & vector multiplication.

Still, i think you're making a good point. So maybe the correct process would be : problem stating, naive "manual" solution, graphical representation and then matrice & vector formalizing ?