Each matrix has a unique* linear transformation associated to it and each linear transformation has a unique* matrix associated to it. Matrix multiplicacion is defined in such a way that it preserves, in matrices, all of the same properties as function composition does in linear transformations. To elaborate, say you have two linear transformations f and g with associated matrices A and B, respectively. If you compose f(g(x)) and then compute its associated matrix, you get AB – the result of multiplying the original matrices. This is why matrix multiplicacion is defined the way it is and I can't figure out for the life of me why nobody told you this in the previous 69 replies.
Math definitions are full of shit that seems to be arbitrary and unintuitive but the reasoning behind it becomes very apparent once you take a deeper look. Another quick example that comes to mind is complex numbers multiplication where you have:
(a,b)*(c,d)=(ac-bd,ad+bc)
This makes no fucking sense at first glance but it turns out that, with this definition, you can write complex numbers as binomials of the form (a+bi) and (c+di) and just multiply them as you usually would, keeping in mind that i^2=-1.
*assuming a specific basis has been chosen but that doesn't really matter since, unless you're a masochist, you'll always be working with the standard base (the e_ij's that anon mentioned here
>>11604604)
