Dear Travis, I basically agree with you, in the sense to fix that what equality means. At this time, equality of vector spaces does not imply equality of associated vectors. I think both choices have good reasons. I am not sure how to collaborate with these issues but I am ready to. Best, Enrique.
El domingo, 27 de diciembre de 2020 a las 2:49:11 UTC+1, Travis Scrimshaw escribió: > We need to be very careful here about what "same" means, and the category > we want to work in is vector spaces with a distinguished basis. By similar > I really meant up to a change-of-basis, and I should have been more > precise. Although perhaps it is a more simple problem than I was thinking > because we know explicitly how to link different bases together (not just > some abstract isomorphism). Thus, the comparison as matrices is the correct > thing to do, provided the two bases are the same. Hence, we just need to > make sure each linear map set up from basis B to basis B'. Therefore, the > bug is in the __call__, which handles the coercion between the elements of > the (distinct) Hom spaces before the _richcmp_ sees the objects. > > Again, let me reiterate that Python == should not always be taken as > equality as you might think of it. However, in this case, if we fix the > aforementioned bug, everything should behave how you expect it to. > > Best, > Travis > > On Thursday, December 24, 2020 at 7:37:10 PM UTC+10 enriqu...@gmail.com > wrote: > >> Not exactly, when one asks if two linear maps are equal it is not the >> same thing to ask if they are similar, I mean equal as maps. In my example, >> if you enter: >> >> K=GF(2) >> V=K^2 >> B=V.basis() >> B1=[vector(K,[i,j]) for i,j in [(1,0),(1,1)]] >> V1=V.subspace_with_basis(B1) >> >> and you ask V==V1 the answer is True despite they are distinct objects in >> Sage as their associated bases are not the same. In my opinion the option >> is correct because as mathematical objects they are the same. >> >> When one defines morphisms with domains V or V1, and same codomain, the >> method keeps the matrix as comparison object. As you say computing the >> Smith form (or Jordan form in case of endomorphisms) is expensive. I am not >> sure if computing the images of the basis elements of either V or V1 is so >> expensive. If it is so, it would be better to avoid an answer for the >> equality, to avoid wrong answers. >> >> This is the source of a._richcmp_?? >> >> def _richcmp_(self, other, op): >> if not isinstance(other, MatrixMorphism) or op not in (op_EQ, op_NE): # >> Generic comparison >> return sage.categories.morphism.Morphism._richcmp_(self, other, op) >> return richcmp(self.matrix(), other.matrix(), op) >> File: >> /usr/local/sage/local/lib/python3.8/site-packages/sage/modules/matrix_morphism.py >> >> Type: method >> >> Best, Enrique. >> >> >> El jueves, 24 de diciembre de 2020 a las 0:41:10 UTC+1, Travis Scrimshaw >> escribió: >> >>> What you're asking for is checking if two matrices defining the linear >>> transformations are similar, which involves computing the Smith normal >>> forms of the matrices. This is computationally expensive. Instead, what we >>> do is == becomes a weaker form of equality that is fast (equivalently has >>> stronger conditions). This is something we do for graphs where we have a >>> separate method is_isomorphic(). Therefore, I think that equal linear maps >>> not being == is okay if they are given by different matrices. >>> >>> That being said, there is a definite issue with the false equality a == >>> c, which because >>> >>> sage: a.parent()(c) >>> Vector space morphism represented by the matrix: >>> [1 0] >>> [0 1] >>> Domain: Vector space of dimension 2 over Finite Field of size 2 >>> Codomain: Vector space of dimension 2 over Finite Field of size 2 >>> >>> which is not the correct map. This is a result of the fact the __call__ >>> does not check that the the (co)domains match and compose with the >>> corresponding coercion maps. >>> >>> Best, >>> Travis >>> >>> >>> On Thursday, December 24, 2020 at 2:07:35 AM UTC+10 enriqu...@gmail.com >>> wrote: >>> >>>> I have posted a bug in hom method for vector spaces in >>>> https://groups.google.com/g/sage-support/c/1VDRFjZePCo/m/sFimwoV2BgAJ >>>> >>>> If one defines linear mappings using hom, distinct morphisms may be >>>> told to be equal and viceversa >>>> >>>> Following a colleague advice (Miguel Marco) I looked at the method >>>> _richcmp_ and it seems that it checks if the matrices of the morphisms are >>>> equal. Since the same vector space can be defined in Sage with distinct >>>> associated bases, this is the reason of the error. >>>> >>>> I guess that equality of domains (and maybe codomains) is the first >>>> condition to be checked and then the equality of the images of self. >>>> >>>> Best, Enrique. >>>> >>> -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/f17e91bb-e767-46e4-adcf-c2e2d6aa6063n%40googlegroups.com.
