Where's the circularity?

What you're saying is fine as an abstract presentation, but I was talking about how students might initially come to learn about matrices, so just introducing column vectors as representing points in 2 and 3 dimensional space and how matrices transform them is fine.

Beginning with the field and vector space axioms might be fine for sophisticated students, but I don't think it would make for an optimal learning experience for most students. We also don't teach kids the Peano axioms before they learn to add and multiply

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