I do not have strong opinions on this issue and I will not be the one to do the work, but I still think "starting at 0" for matrices is more natural. Of course, the more important thing is to have as many results as possible stated for arbitrary finite sets (or even arbitrary sets for some results?). Actually, I think starting at 0 is preferable even more after reading Mario's arguments, and the shift from literature is not that important (I think it won't even confuse beginners, since this is such a minor variation.)
As Norm said, it depends on the intended audience. I mentioned this question a few times when discussing other topics, and said it was bound to show up again and again, since many decisions depend on this question. Of course it's better to fulfill the needs of different audiences, but it's not always possible. You don't write a textbook the same way you write a reference treatise (admittedly, with electronic material, one might have more flexibility). Personally, I think it's time for Norm to realize the grandeur of what he created ;-) and that it's closer to Bourbaki's Elements than to your usual textbook. Benoît On Saturday, September 5, 2020 at 8:43:40 AM UTC+2 Alexander van der Vekens wrote: > As proposed by FL in > https://groups.google.com/g/metamath/c/n5g69qfwBmE/m/McJAgtdSAgAJ, the > additional assumption that the finite index sets are totally orderer should > be sufficient to express and prove the Laplace expansion. Regarding the > expression ` -1 ^ ( i + j ) ` , a special concept of parity must be defined > for (finite) totally ordered sets... > > On Wednesday, September 2, 2020 at 7:02:23 PM UTC+2 Thierry Arnoux wrote: > >> On 02/09/2020 23:25, 'fl' via Metamath wrote: >> >> The best to do is redefine the <" ...>" operator so that it takes (1 ... >> N) as its set of indices and then fix up all the proofs referring to the >> definition. >> You should have only one definition for matrices, tuples and words since >> all that is the same story. Or at most two: one with a abstract finite >> set of indices and another one with (1... N). >> >> Like mentioned by Norm in the original thread about index start for >> words, I'm afraid that would be a huge work, there are already hundreds of >> theorems making use of that range. >> That's why I would like to ask the same about matrices now, before >> writing more theorems within whichever convention. >> >> Of course we shall try to use arbitrary sets whenever possible... which >> leads me to the other point: >> >> On the other hand, does any one have any idea or suggestion about how I >>> could have expressed the Laplace expansion without integer indices, on any >>> finite set index? >>> >> >> The laplace expansion of the determinant (just one level of expansion, >> not recursive) requires a choice of row to expand over (which is an element >> of the index set), and then it's an unordered finite sum. No integers >> needed. >> >> Yes, but what is the factor you apply to each submatrix determinant ? The >> textbooks all have ` -1 ^ ( i + j ) ` , where i and j are *integer* >> indices. Of course, start index does not matter, but where do those numbers >> come from when working with arbitrary sets? >> >> _ >> Thierry >> > -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/metamath/cf9c496f-fbda-42b2-ba01-f1f17a6fee53n%40googlegroups.com.
