As I am studying linear algebra I can't help but think that the method of arranging coefficients in an array like structure, what we now call matrices came about in an arbitrary fashion. Further the operations defined upon them, though clearly motivated by the need to encapsulate certain information, do seem in some ways to be a product of this arrangement. Is it really an arbitrary arrangement and if it were not for historical reasons, could we have had matrices structurally different and with different semantics? Please expound if I am mistaken.
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When you think to the operations, with an "s", in fact there is no problem with addition and multiplication by a real; thus, you must find "akward" only the product with the assymmetrical rule "line of the first times columns of the second". Do I summarize correctly? – Jean Marie Oct 29 '16 at 20:58
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This might interest you : http://math.stackexchange.com/questions/271927/why-historically-do-we-multiply-matrices-as-we-do – Arnaud D. Oct 29 '16 at 22:10
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That question is somewhat related, but not quite there. I am asking whether the entire arrangement is arbitrary or not. Imagine if we could develop an entirely new system by arranging coefficients in a circle and then defining operations on them, keeping the essence of the definitions the same, but only changing the semantics. Could it be done? – SaitamaSensei Oct 30 '16 at 03:58