Talk:Multilinear algebra

New To Advanced Math

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)

Fix it then. Anyone can edit, you know. 70.186.172.75 (talk) —Preceding comment was added at 13:05, 22 April 2008 (UTC)

Adding unencyclopedic tone tag. For example, "there was probably no going back in purely mathematical terms." and "In the category-theoretic jargon, everything is entirely natural." These read more like the history section of a pure math paper than an encyclopedia article. 96.40.171.147 (talk) 16:12, 11 June 2012 (UTC)

No such thing as new or topicx or conclux or not, say any. — Preceding unsigned comment added by Lyhendi (talk • contribs) 19:52, 11 September 2018 (UTC)

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

I recommend deletion...

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)

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)

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)

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)

First of all, my qualification for saying this is that I have a minor in mathematics from the University of Arizona in 2003 and I have continued studying mathematics and physics since that time so I believe I am at least qualified to have an opinion on the quality of the article.

I think the article is a reasonable description of a complex mathematical field of study that requires most of a semester to master (or at least begin to master) for students who have had a couple of years of calculus plus linear algebra with a course on differential geometry as an added plus. Of course the more background, the easier these concepts are to understand.

With that being said, (and I hope I don't sound too pontifical here) expecting an encyclopedia description of the subject to take the place of the course and expecting the article to completely describe the subject is simply expecting too much. Of course the article could be expanded and be more descriptive but it is still an article in an encyclopedia and not a complete course. Just my opinion. Thank you Historyman1phy (talk) 00:01, 16 February 2015 (UTC) — Preceding unsigned comment added by 98.172.22.61 (talk) 23:39, 15 February 2015 (UTC)

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)

Algebraic topology

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 {{Cite book |first=Nicolas |last=Bourbaki |url=http://worldcat.org/oclc/25747293 |title=Algebre. |date=1962 |publisher=Hermann |oclc=25747293}} 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)

Is "moduli" a typo?

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)