Talk:Multilinear algebra

message: Hi;
I'm trying desperately to understand many of these advanced principals of mathematics, such as multilinear algebra, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this?
beno 26 Jan 2006
Yeah, there should be one of those wikitextbooks on this --tcamps42
Clearly, this article was mostly written by a physicist who does not understand mathematics. --anon  —Preceding unsigned comment added by 68.197.9.185 (talk) 03:27, 3 January 2008 (UTC)
replies: text: Fix it then. Anyone can edit, you know. 70.186.172.75 (talk)  —Preceding comment was added at 13:05, 22 April 2008 (UTC)

text: What happened to using an s for plurals? What is conclux supposed to be anyways? 2abc3 (talk) 01:06, 11 April 2024 (UTC)
thread_title: New To Advanced Math

message: I recommend that this article be deleted.  Its scope includes the articles on tensors and exterior algebra.  Its content seems to be a commentary on theoretical mathematics with a list of other articles.  I understand this theory, but I could not find a useful approach that builds on what is provided. Prof McCarthy (talk) 04:49, 25 March 2014 (UTC)
replies: text: I wouldn't, because we can let that others have  the chance to read the ideas in another perspective so they nurture their understandings. kmath (talk) 02:40, 30 March 2014 (UTC)

text: My initial reaction was that it seems to simply try give a new name to the topics you list, and that deletion would have been appropriate. However, I was surprised to see the number of books that use the phrase multilinear algebra. This would seem to argue for expansion and clarification rather than deletion, though with what content I cannot say. It seems to include more than the two topics mentioned, even though all of them can probably be treated in terms of tensor algebra. Thus, bilinear maps might not be immediately recognized as tensors by someone, but are considered part of multilinear algebra. Similarly, the exterior algebra can presumably be fully treated by tensor algebra, but uses a different notation and implicit restriction to the antisymmetric part of every product. Perhaps "multilinear algebra" is about showing the equivalence of the approaches? —Quondum 18:11, 30 March 2014 (UTC)

text: I think the right understanding is that "multlinear algebra" is to "tensors, exterior products, grassmannians, etc" as "linear algebra" is to "vectors, linear transformations, matrices, etc."  So to improve this article, the linear algebra article would be a good template.  Put another way I don't think there is another single article that the term "multilinear algebra" could redirect to without missing out on context (and what is the purpose of an encyclopedia if not to provide context?) 216.15.65.100 (talk) 20:04, 10 April 2015 (UTC)
replies: text: The topic of multilinear algebra is important and I believe it can be presented in a way that the beginner can understand.  I will work on it a little more. Prof McCarthy (talk) 16:08, 1 April 2014 (UTC)
thread_title:  I recommend deletion... 

message: The following was moved here for discussion:
===Use in algebraic topology===
Around the middle of the 20th century, tensors were reformulated more abstractly. The Bourbaki group's treatise Multilinear Algebra was especially influential—in fact, the term multilinear algebra may have originated there.<ref  ref>
One reason at the time was a new area of application, homological algebra. The development of algebraic topology during the 1940s gave additional incentive for the development of a purely algebraic treatment of the tensor product. The computation of the homology groups of the product of two topological spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion (see Künneth theorem). The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined.
The material to organize was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to de Rham cohomology, as well as more elementary ideas such as the wedge product that generalizes the cross product.
The resulting rather severe write-up of the topic, by Bourbaki, entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups), and instead, applied a novel approach using category theory, with the Lie group approach viewed as a separate matter. Since this leads to a much cleaner treatment, there was probably no going back in purely mathematical terms. (Strictly, the universal property approach was invoked; this is somewhat more general than category theory, and the relationship between the two as alternate ways was also being clarified, at the same time.)
In fact, what was accomplished is essentially an explanation of why tensor spaces are the necessary constructs to convert multilinear issues to linear problems. There is no geometric intuition in this purely algebraic approach.
By re-expressing problems in terms of multilinear algebra, there is a clear and well-defined "best solution": the constraints the solution exerts are exactly those needed in practice. In general there is no need to invoke any ad hoc construction, geometric idea, or recourse to co-ordinate systems. In the category-theoretic jargon, everything is entirely natural.
===Comments===
The Bourbaki reference was moved up with citation of Elements. This otherwise unreferenced, informal section requires editing. Rgdboer (talk) 23:11, 8 May 2023 (UTC)
thread_title:  Algebraic topology 

message: The Bourbaki reference was moved up with citation of Elements. This otherwise unreferenced, informal section requires editing. Rgdboer (talk) 23:11, 8 May 2023 (UTC)
thread_title: Comments

message: I honestly can't tell, since the plural for model is models, but it may mean something different. Since I can't tell, can someone with more info inform me? 2abc3 (talk) 18:16, 3 March 2024 (UTC)
thread_title:  Is "moduli" a typo?