[nh23423fefe]

But the question was about deriving the multiplication rule. I said you could derive it from systems of equations directly and gave a proof.

> and how matrices transform them is fine

this is circular. You are introducing/assuming the multiplication rule right here. You can't then derive it

[griffzhowl]

Yes, I was assuming it, not deriving it, so there's no circularity.

I realise though that I was answering more "How does it work?" (application) rather than "Why does it work?" (derivation)

For the latter, something involving sets of linear equations is probably best, as you initially said