I think the idea is to recontextualize the coordinate system of your object as not being the global coordinate system in the first place. The matrix defines the

I think the idea is to recontextualize the coordinate system of your object as not being the global coordinate system in the first place. The matrix defines the orientation of your object relative to the global coordinate system. The question of "how to rotate something" rests on operational semantics. The question of "how is this related to that" rests on denotational semantics, and is much more general. The beauty of linear algebra is how closely the two are related: every linear transformation (the relationship) gives a matrix given a choice of basis, and matrices have a direct algorithm for transforming coordinate vectors.