I’m trying to understand the fundamental reason behind why geometric transformations in computer graphics are usually represented as a matrix multiplied by a vector.

Is it because a matrix is essentially the only mathematical tool that can uniformly describe how a whole set of points moves from (x, y, z) to (x', y', z')? Or are there deeper theoretical or practical reasons that make matrices the standard choice?

I’m not asking about API usage, but purely about the underlying mathematical justification.