Talk:Exterior algebra/Archive 2

Archive 1

Archive 2

Archive 3

Elements' degrees don't just add

They might as well become zero. I'm not sure it's worth mentioning in the lede, though — Kallikanzarid^talk 21:25, 20 January 2011 (UTC)

I think a tiny error of omission is probably admissible here. But one counterintuitive aspect of the exterior algebra that perhaps should be pointed out is that it has zero divisors, and this might be a good place in the text to compare and contrast with the case of ordinary polynomials. Sławomir Biały (talk) 17:30, 21 January 2011 (UTC)

Second paragraph

I have removed the following paragraph from the lead of the article, as it seems like dead weight. Any thoughts about whether and how this content should be re-included? Sławomir Biały (talk) 16:21, 20 January 2011 (UTC)

In applications to linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a matrix. For instance, it is well-known that the magnitude of the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix. This suggests that the determinant can be defined in terms of the exterior product of the column vectors. Likewise, the k×k minors of a matrix can be defined by looking at the exterior products of column vectors chosen k at a time. These ideas can be extended not just to matrices but to linear transformations as well: the magnitude of the determinant of a linear transformation is the factor by which it scales the volume of any given reference parallelotope. So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power. The action of a transformation on the lesser exterior powers gives a basis-independent way to talk about the minors of the transformation.

Whatever we can say about determinants, minors, parallelopipeds, and column vectors (and also rank) in the lede is far preferable to blades, functors, transformation of categories, and universal constructions. Tkuvho (talk) 19:26, 20 January 2011 (UTC)

we are constrained by a need to summarize the article in the lead, and a large part of that is dedicated to these issues. Sławomir Biały (talk) 20:29, 20 January 2011 (UTC)

I suggest moving its content to the main body, maybe with other examples like vector calculus operators. It is useful in explaining why exterior product almost completely supplanted determinants (I remember it being discussed at mathoverflow). — Kallikanzarid^talk 19:30, 20 January 2011 (UTC)

For now, I have put the paragraph into the Applications section. I don't claim that this is the perfect place for it, and it could use some expansion and tweaks either way. Sławomir Biały (talk) 21:31, 22 January 2011 (UTC)

comments on lead section

Per Sławomir's request, I'm giving my 2c about the lead section here: first of all, I have made the experience (at the failed FAC of vector space) that some editors wish to have a lead section, which explains the concept (at least partially) to the layman. My personal response to this (was and) is, that the lead section cannot serve this purpose, since it is deemed to be succinct and should should summarize the article. To the best reasonable/possible extent, an introduction section should provide detailed motivation and explanation of preliminaries. In the lead, a little bit more weight might be given to basic items, but generally I believe the lead should be a proportional copy of the article, which means each section is covered in the lead. This is my personal view on this, but proved helpful so far.

As for this particular lead section (this version):

• Generally speaking, I find the lead extremely long. Too long, that is.
• Sometimes it is too detailed ("meaning that u ∧ u = 0 for all vectors u, or equivalently[2] u ∧ v = −v ∧ u for all vectors u and v" should be shrunk to one variant, I believe the latter is more common and easier to digest?)
• I find the emphasis on the relation to cross products not so helpful. At least I never think of the cross product when thinking of the wedge product.
• Some of the mentioned points seem to be less relevant: associativity, gradedness, simple elements
• I disagree with "even if in many geometrical applications the only objects of interest are the simple elements" (think about integration on a manifold)
• What specifically does "while it is possible to describe the exterior product geometrically in terms of volumes, the exterior algebra provides a much more mathematically satisfactory definition of that product" mean?
• "Differential forms are, roughly, things that generalize the differentials appearing in line integrals to objects can be integrated over curves, surfaces, and higher dimensions manifolds." (emphasis added) is choppy wording. However, the point is good I believe, but should/could be explained more smoothly by alluding to an infinitesimal parallelogram, which gives rise to integration.
• "The exterior algebra also has many desirable algebraic properties that make it a convenient tool in algebra itself. " -- words like desirable are often problematic and/or superfluous. I would remove this whole sentence.
• "The exterior algebra can be associated to any given vector space V over a field K. It is a unital associative algebra denoted by Λ(V)" we already had this bit of information before.
• Another first-sight comment: the applications are surprisingly short. On the other hand, the formal properties are very detailed. Don't go any further in that direction, would be unbalanced IMO.

Maybe a way to rewrite the lead would be this (wording and many other things to be improved!)

[picture here] In maths, the wedge product is a way to multiply vectors with one another. A guiding example is the wedge product u \wedge v of two planar Euclidean vectors, that is to say, arrows in the plane. This wedge product can be thought of as the area of the parallelogram defined by u and v. However, the area is understood in a signed (or oriented) way: switching the order of the factors results in alternating the sign of the wedge product:

$u\wedge v=-v\wedge u.$

[If you really want this: The wedge product is also generalizing the triple product and the cross product. ] The vectors of some given space, together with their wedge products build an algebraic structure where both addition and multiplication is possible. It is called exterior algebra. In the language of linear algebra, [put this here to tell the reader that he cannot understand the following if he does not know v.sp.!] the exterior algebra Λ(V) of some vector space V is defined as the quotient of the tensor algebra of V by anti-commutativity relation mentioned above. Thus, in a universal sense, it is the freest [?too technical?] possible algebra containing V and satisfying the anti-commutativity relations. [So far we have told what this is about, now some properties. Since the technical properties are quite boring, I would not go into much detail]

Given a linear map between two vector spaces, the resulting exterior algebras are also connected by a map. Moreover, the exterior algebra has a strong flavor [??] of duality: not only is it an algebra, but also a coalgebra. [maybe give a motivating geometric explanation for the Hopf algebra structure?] The exterior algebra goes back to the work of Grassmann, who [did this and that with it]. The modern definition places it in the heart of multilinear algebra, which is concerned with linear maps in several variables. It applications include [...]. Many of these applications require the concept of exterior algebra to be generalized to more general structures than vector spaces, such as vector bundles and modules.

Jakob.scholbach (talk) 19:24, 21 January 2011 (UTC)

Yes, it's too much long. It should be more like the history section in length: three or four short paragraphs which summarise the topic and are accessible. Long paragraphs are daunting and inaccessible at the best of times. When dealing with an advanced maths topic they can be impenetrable. Again look at the paragraphs in the history section or the lede of any good mathematics article.

The lede doesn't need to summarise all the article, just its main points. This is especially true of mathematics articles which can contain both [more] elementary and advanced topics in the same article, as well as digressions from the main topic which may be of only specialist interest. Jakobs version above is much closer to the proper length and content. I would add more detail on the cross product but I don't see much more that needs adding.--JohnBlackburne^words_deeds 21:32, 21 January 2011 (UTC)

Some comments: an earlier revision of the lead, perhaps closer to Jakob's suggestion, was this, although there were complaints that this was too high-brow for all but the most mathematically sophisticated readers. I disagree with John's remark that long paragraphs are less accessible. See my comment above about mathematicians often being too terse for a general audience. Where we could just say that the exterior algebra is a graded associative algebra, because of the need to avoid using jargon, instead we need to spend four or five sentences explaining what this means and why its useful. A part of explaining this includes saying that the exterior algebra has non-simple elements. That said, I do feel that the lead as it currently stands is too long, but I don't really see how to avoid it. We want it to be inviting to non-mathematical readers, and so far it seems to me to be successful at this. One of the things that we could try is a different kind of "Introduction" section that includes the text of the current lead. But readers were complaining that they weren't making it past the first paragraph to get to the existing "Motivation" section. Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)

(@Jakob) I have made some edits in response to your comments. I am wondering if you would prefer to go back to this version of the lead? If so, what would you suggest doing with the new content that I have added? Here I have responded to your points:

• Generally speaking, I find the lead extremely long. Too long, that is.
  • I also find it too long, but I don't see how to carefully explain the topic and have a short lead. Perhaps a separate "Introduction" section would help, but in mathematics articles, these are often useless and badly written. Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
    • Well, the quality of other articles cannot quite be an argument. Many articles are just in general crappy. In group (mathematics) and vector space we tried (with mixed success, according to FAC) to give an introductory section. Now exterior algebra is certainly more advanced than the two, but the likely readership for this article will also be more educated than there. Jakob.scholbach (talk) 19:13, 22 January 2011 (UTC)
      • I guess I'm trying to say that I think I lack a useful model on which to base such a section. I thought that the previous version of the lead was ideal, but some readers (who even claimed to have university-level mathematics background) found it so impenetrable that they apparently weren't interested in getting as far as the table of contents. So I would be happy to have an "Overview" or "Introduction" section, but then it isn't clear to me how to write the lead. After all, this is a substantially more advanced topic than vector space or group, but yet also one that apparently has substantial interest outside of mathematics (physics, engineering, computer graphics). With the comparative sophistication of the topic, and the potential breadth of readership, how should one structure the lead to make it more inviting than the old one? Do you know of any examples? Sławomir Biały (talk) 20:51, 22 January 2011 (UTC)
• Sometimes it is too detailed ("meaning that u ∧ u = 0 for all vectors u, or equivalently[2] u ∧ v = −v ∧ u for all vectors u and v" should be shrunk to one variant, I believe the latter is more common and easier to digest?)
  • Done. Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
• I find the emphasis on the relation to cross products not so helpful. At least I never think of the cross product when thinking of the wedge product.
  • It's likely to be something that the less sophisticated readers are already familiar with. Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
    • The cross product is an example of a determinant and in this sense already covered. This is the connection that I would emphasize. Comparing the cross product to exterior products forces you (as the current version acknowledges) to first point out the similarities, then the differences. Jakob.scholbach (talk) 19:13, 22 January 2011 (UTC)
      • It should probably be possible to rewrite the first paragraph to avoid discussing cross products. Is it worth keeping the image then? Other editors seem to have found this image particularly helpful. Sławomir Biały (talk) 20:58, 22 January 2011 (UTC)
      • I've done this. The first paragraph is more streamlined, but I'm pretty sure it's not better from an accessibility point of view. Any thoughts? Sławomir Biały (talk) 21:27, 22 January 2011 (UTC)
• Some of the mentioned points seem to be less relevant: associativity, gradedness, simple elements
  • I definitely agree, but I think they are likely to help someone who has never encountered an associative algebra appreciate the structure. (For instance, often beginners think that every element is simple, which is not true and misses nuances in how the product is defined.) Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
    • I think someone who has never seen the word "associative", "associative algebra" (or has not understood the notion of ass. algebra) will simply wonder why this information is there. On the other hand, anyone who has a little bit of algebraic routine will (rightly) suppose that the associativity is trivial. So I think this is dispensable in the lead. Gradedness is intuitively clear and again I think a secondary piece of information. Jakob.scholbach (talk) 19:13, 22 January 2011 (UTC)
      • You could be right. It doesn't seem to be a major point either way. Sławomir Biały (talk) 20:39, 22 January 2011 (UTC)
      • I've trimmed the verbosity a bit. I think the lead should link associative algebra (the old version linked unital associative algebra), and should say a little bit about the graded structure because this appears in the table of contents. Sławomir Biały (talk) 21:27, 22 January 2011 (UTC)
• I disagree with "even if in many geometrical applications the only objects of interest are the simple elements" (think about integration on a manifold)
  • I have replaced the word "many" with the word "some". Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
    • Still unclear to me. In what domain do you explicitly not (want to) use the additive structure in the ext. algebra? Jakob.scholbach (talk) 19:13, 22 January 2011 (UTC)
      • See my reply to the next point. Sławomir Biały (talk) 20:39, 22 January 2011 (UTC)
• What specifically does "while it is possible to describe the exterior product geometrically in terms of volumes, the exterior algebra provides a much more mathematically satisfactory definition of that product" mean?
  • The intention here is to impress that it is possible to define, e.g., products of vectors by appealing to geometry, really the power of the algebra is that the product is defined without appealing to geometry. Thus, for instance, we can use it as a practical tool for doing real calculations. There is something of this in the next sentence as well. Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
    • Really? How do you geoemtrically define the wedge product of two vectors in R^5? Of course you can write down the formula per pedes, but I think nobody does this. Hence pointing out that one could do what nobody does seems to go in the wrong direction. Jakob.scholbach (talk) 19:13, 22 January 2011 (UTC)
      • Well that's sort of the point. We have defined the exterior product (of vectors at least) in terms of volumes in the first paragraph. But that's clearly not satisfactory from anything but an intuitive point of view. Sławomir Biały (talk) 20:39, 22 January 2011 (UTC)
• "Differential forms are, roughly, things that generalize the differentials appearing in line integrals to objects can be integrated over curves, surfaces, and higher dimensions manifolds." (emphasis added) is choppy wording. However, the point is good I believe, but should/could be explained more smoothly by alluding to an infinitesimal parallelogram, which gives rise to integration.
  • Done (or at least attempted). Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
• "The exterior algebra also has many desirable algebraic properties that make it a convenient tool in algebra itself. " -- words like desirable are often problematic and/or superfluous. I would remove this whole sentence.
  • I got rid of the word "desirable". Amid other edits, though, a segue still seemed to be needed, so I left the rest of the sentence in place. Sławomir Biały (talk) 14:31, 22 January 2011 (UTC)
    • As a non-expert, I want you to just tell me outright when something possesses features that are desirable, unusual, efficient, convenient, useful, etc. These sorts of adjectives signal to me that this type of algebra is important. WhatamIdoing (talk) 17:11, 23 January 2011 (UTC)
• "The exterior algebra can be associated to any given vector space V over a field K. It is a unital associative algebra denoted by Λ(V)" we already had this bit of information before.
  • Killed. Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)
• Another first-sight comment: the applications are surprisingly short. On the other hand, the formal properties are very detailed. Don't go any further in that direction, would be unbalanced IMO.
  • No comment. Sławomir Biały (talk) 14:04, 22 January 2011 (UTC)

All in all I'm sceptical about the idea to teach the topic in question in the lead section. Maybe this is just a question of habit. On the other hand, I feel a well-done introduction section has more space to carefully lay down the basics. If the lead is written in a way that enables people to scroll down just after the table of contents, this does just the same job.

As for the maths, maybe another way of presenting the exterior algebra is this: link the highest exterior power to determinants (as it is done now). Then link intermediate ext. powers to minors. Jakob.scholbach (talk) 19:13, 22 January 2011 (UTC)

You may have noticed that all references to minors were recently eliminated in a massive deletion of a long-established paragraph. The latter was replaced by Bourbaki-style intimidation concerning universal constructions and equivalence of categories. All this stands in direct opposition to lede guidelines, both general and those tailored to mathematics pages. Tkuvho (talk) 19:56, 22 January 2011 (UTC)

You are wrong on both counts. The "long standing paragraph" whose removal you are objecting to is one that I wrote just a few days ago. In contrast, what you claim as a "Bourbaki"-style content on universal constructions has been part of the lead of the article almost word-for-word for years. Sławomir Biały (talk) 20:32, 22 January 2011 (UTC)

proposal to delete Bourbaki=style paragraph and restore minors, ranks, cross products

I propose the deletion of the last paragraph of the introduction containing an obscure discussion of equivalences of categories and natural constructions. I also propose to restore the material that has been deleted on minors, ranks, cross products, and other items that can help connect the reader's previous knowledge to the subject of this page. I believe such changes are consistent with our guidelines for lede pages. Tkuvho (talk) 21:08, 22 January 2011 (UTC)

The new third paragraph has been particularly praised by a novice reader, and you seem to be the only one claiming that it is excessively Bourbakist. Refer to RobHar's reply to your same complaint at WT:WPM. Moreover, the third paragraph is the only paragraph that refers to any applications, and it summarizes a good 50% of the article. Removing this paragraph would clearly violate WP:LEAD, since the primary purpose of the lead is to give a summary of the article that is proportional to the coverage in the article. Sławomir Biały (talk) 21:14, 22 January 2011 (UTC)

Also, there is nothing in the lead about natural transformations or equivalences of categories. We mention briefly that it is a universal construction, which seems to be appropriate emphasis given that there is a lengthy subsection dedicated to this approach. We mention that the algebra can be constructed in terms of known objects like tensors. This seems equally appropriate. There is a sentence on functoriality because there is a section on the functorial properties. There is a sentence about duality because there is a section on duality. It's really hard for me to see why mentioning these things in the lead would be so controversial for you. The last paragraph is the only one that seems to be following the letter and spirit of WP:LEAD fairly closely. Sławomir Biały (talk) 21:20, 22 January 2011 (UTC)

I was referring to "universal constructions" (delete "natural"). This kind of talk is not appropriate in a lede, which should be accessible to college juniors. The applications should certainly be kept, but not equivalence of categories and universal constructions. Your novice reader must be a particularly brilliant one, as only a few days ago he did not know what an exterior algebra is, and now you interpret him as being excited about equivalence of categories :) Incidentally, last time I read the lede it still said nothing about the de Rham theory which is the main application you seem to have in mind. Why talk around it and not mention it? At any rate, most of this discussion should probably be at exterior differential complex and not here. We are trying to explain exterior algebra, not vector bundle of exterior algebras. As the next step, this can certainly be mentioned in the body of the article, but going to this level of detail in the lede is inappropriate. By no reasonable criterion can one claim that this is accessible "to a general audience" as WP:LEAD requires. Tkuvho (talk) 08:00, 23 January 2011 (UTC)

A basic rule of thumb is that anything that appears in the table of contents should be mentioned in the lead. Universal constructions are there because they appear in the table of contents. Equivalence of categories are not mentioned at all in the lead (or the article). Sławomir Biały (talk) 13:06, 23 January 2011 (UTC)

I believe it is this very "rule of thumb" that we are discussing. I think that the lede should be accessible to college juniors. Therefore mentioning advanced topics such as "functors" are not helpful. "Equivalence of categories" has since been deleted, as you point out, but we still have "functors". Also, ranks and minors should be restored. Incidentally, I appreciate the tremendous amount of work you put into the page recently. I think trying to dumb down the lede to an accessible level will make the page even better. Tkuvho (talk) 13:16, 23 January 2011 (UTC)

At no point in its history did the lead ever mention "equivalence of categories". The discussion of ranks and minors was too long for the lead: at any rate it was certainly highly disproportionate to its coverage in the article (referring again to WP:LEAD). For now I have moved it to the applications section. Personally, I think that while it is a good idea to make the lead as accessible as possible, that does not include "dumbing down" by not mentioning more advanced content (WP:MTAA has some insight on this very question). The main purpose of the lead is to summarize the article. Sławomir Biały (talk) 13:26, 23 January 2011 (UTC)

The very page you referred me to starts with the following phrase: "Articles in Wikipedia should be understandable to the widest possible audience. For most articles, this means understandable to a general audience." I don't think functors make this understandable to a general audience, which in this case can be reasonably interpreted as college juniors with a math orientation. Similarly, talk about "universal constructions" is useful to someone with considerable experience in dealing with such algebraic objects. From personal teaching experience I know that it is very hard to explain the idea of "naturalness" to students, it only comes with a lot of practice. Its presence in the lede would only confuse a typical junior. When "summarizing the article" comes in conflict with "making it accessible", the latter directive bears more weight. As one of the editors pointed out at WPM recently, the lede should not be a scientific abstract. Tkuvho (talk) 13:33, 23 January 2011 (UTC)

See the part about not "dumbing down" the article. WP:MTAA explicitly says that the article should not be made accessible at the expense of other aspects of the content. In other words, it is the secondary consideration here (which is already suggested by the fact that MTAA is an essay which assumes a lower priority than LEAD which is a guideline). Finally, I don't think anyone can credibly claim that any part of the lead as it currently stands is at all like a scientific abstract. Sławomir Biały (talk) 13:52, 23 January 2011 (UTC)

If that's the case, you can't credibly claim that the lede "summarizes" the article? What's the point of keeping the "universal construction" and "functor" that will only scare away a majority of readers? Tkuvho (talk) 13:56, 23 January 2011 (UTC)

We make an effort to explain all (or most of) the jargon. Scientific abstracts never do this. Sławomir Biały (talk) 14:04, 23 January 2011 (UTC)

Explain everything except for "module coefficients", that is. Tkuvho (talk) 14:33, 23 January 2011 (UTC)

Well, that's not a phrase that appears in the lead, and I'm not even sure what it is supposed to mean. We should say that the exterior algebra makes sense for modules. We do this by first suggesting a special case, and then articulating the general one. That seems to be a fairly common expository style. There is a related discussion below. 14:41, 23 January 2011 (UTC)

K's thread

1. Define Bourbakism,
2. The notion of an algebra being the largest one is intuitive. — Kallikanzarid^talk 08:52, 23 January 2011 (UTC)

I am not gonna jump into these current discussion, but I wonder why reference to the cross product has been deleted from the lead section? IMHO, this was an essential part of accessibility. Suggest: The magnitude[1] of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which can also be computed using the cross product. Also like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −v ∧ u for all vectors u and v. Furthermore, the vector names should match that of the diagram, i.e., use u and v in the diagram as well (or a and b in the text). For a novice to the topic I imagine that it may be hard to understand when you say "as an oriented plane element: that is a family of parallelograms". Why not simply say "as a family of parallelograms"? I am fine with the following two paragraphs, which extend the concept and outline its usability and usefulness. Nageh (talk) 10:43, 23 January 2011 (UTC)

I went WP:BOLD. If you disagree with some change feel free to revert. Nageh (talk) 12:00, 23 January 2011 (UTC)

I obviously agree with your sentiment that connection to elementary concepts such as cross product should be restored, and similarly for rank, minor, etc. What is behind this deletions is a misguided educational philosophy as I argue here; feel free to contribute your thoughts. Tkuvho (talk) 12:02, 23 January 2011 (UTC)

Tkuvho, stop this ad hominem nonsense. If you have something concrete to say, say it. — Kallikanzarid^talk 12:42, 23 January 2011 (UTC)

I don't have any problems with them — Kallikanzarid^talk 12:42, 23 January 2011 (UTC)

Connection with determinants

I wrote something about connection between wedge product and determinants: User:Kallikanzarid/Sandbox#Determinants. I haven't sourced it or checked it for correctness really hard, I also don't think it will fit anywhere in the article verbatim. I encourage everyone to salvage it and use in Motivation and Examples sections, though :) — Kallikanzarid^talk 09:52, 23 January 2011 (UTC)

Can someone explain why the full exterior algebra is useful (to a layman)?

I'm referring to this removal, which is to a rather controversial bit of text that I confess to adding. My goal was to explain in simple terms why one would be interested in this huge algebra even if it was not immediately motivated from elementary geometric considerations. One answer is that the product in the algebra can be manipulated in terms of a simple set of rules, and that obviously facilitates even the most basic calculations that use the exterior product. This was my intention in writing the above, although it was removed as a "questionable POV". Based on discussion with Jakob, I see that he was also confused about my intentions. Moreover, I think the lead reads better without the two sentences. So I'd like to poll the folks here whether they think it would be helpful to include a sentence that conveys the usefulness of the exterior algebra to laymen, and if so what a better way to do this is. Sławomir Biały (talk) 13:18, 23 January 2011 (UTC)

Thanks for your constructive attitude. A particular problem with the sentences was the claim that only the simple ones are useful but not linear combinations. This is manifestly not the case as the standard treatment of Green's theorem involves a linear combination. Of course, any 1-vector is simple with respect to a suitable base, but at any rate what appears in the standard statement of Green's theorem is not a "blade" but a linear combination. As far as the usefulness of the exterior algebra, it certainly comes from de Rham theory and related developments. As an algebraic object in its own right I am not sure it is that useful, but maybe there are other applications. What's more useful is the Clifford algebra. Tkuvho (talk) 13:26, 23 January 2011 (UTC)

I had hoped that the word "some" in that sentence ("Even if the only objects of interest in some geometrical applications are the simple elements...") would not have given anyone the impression that this is true in general. I certainly don't feel that only blades are important. Sławomir Biały (talk) 13:29, 23 January 2011 (UTC)

This still doesn't really address my question. There is no question that the exterior algebra is useful because of its connection with differential forms. The issue here is how to convey the need to pass from a naive geometrical point of view (where one is perhaps interested in volumes and other fairly concrete objects) to a more algebraic point of view, where one has the full exterior algebra. That is, how do we motivate introducing the full exterior algebra? Sławomir Biały (talk) 14:31, 23 January 2011 (UTC)

The way I do it in teaching is via complex projective space, where there is an unavoidable need to work with 2-forms. Tkuvho (talk) 14:51, 23 January 2011 (UTC)

That's not likely to be much help to the layman. Maybe there is a different editor who has a more useful point of view? Sławomir Biały (talk) 15:00, 23 January 2011 (UTC)

I disagree that the exterior algebra is not useful as "an algebraic object in its own right". It is one of the most fundamental examples of differential graded algebras. A tremendous amount of homological algebra can be done with DGAs alone, and there are aspects of homological algebra which become simpler. Ozob (talk) 19:49, 23 January 2011 (UTC)

I don't think anyone here would dispute that perspective. What was needed though was a way to motivate the transition from naive geometrical considerations to purely algebraic ones. I think I have found a way to express this same idea in a less objectionable way. diff Sławomir Biały (talk) 20:00, 23 January 2011 (UTC)

exterior algebra applied to vector fields

The phrase "The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions" is misleading. I already mentioned this above, but the phrase has not been modified yet. The phrase makes it appear as if the exterior algebra construction is being applied to the vector space formed of such vector fields, etc. (Particularly in the context of the "universal construction", it sounds as if we are now applying the construction to this infinite dimensional vector space.) This is of course not the case, as has been discussed both here and WPM. Rather, the exterior algebra construction is being applied pointwise to each (co)tangent space. In other words, we have a bundle of exterior algebras, rather than the exterior algebra of the space of sections of the (co)tangent bundle. Tkuvho (talk) 14:09, 23 January 2011 (UTC)

I think you have been corrected both here and at WPM. But let's do it again. The space of sections of the cotangent bundle is a $C^{\infty }$ module, and the space of differential forms is the exterior algebra of this module. Or, if you think pointwise, then it is the universal construction applied in the category of bundles. But in any event, you seem to be taking very specific information from a sentence that is clearly meant to be a bit vague. It's purpose is to segue into mentioning modules over commutative rings. The novice is likely to be familiar with functions and vector fields, and this should motivate why one should care about the more general cases, or at least to convey something of the flavor of these generalizations. Sławomir Biały (talk) 14:20, 23 January 2011 (UTC)

The "deliberate vagueness" you are referring to is precisely the problem. At the point where the sentence occurs, nothing has been said about more general module coefficients, and obviously nothing can be said without hopelessly confusing the reader. The discussion of bundles of exterior algebras (or, if you like, exterior algebras over module coefficients, etc.) should be left for the last section in the page, along with "functors" and "universal constructions", while rank and minor should be restored, and may we all make an effort to overcome latent Bourbakism. Tkuvho (talk) 14:28, 23 January 2011 (UTC)

Can we please focus on one issue per thread? The sentence is there because many readers may not be know or be comfortable with the language of modules and rings, whereas they are more likely to see functions and vector fields as generalizations of the notion of vector at a point. This is an example of the concentric style of encyclopedia writing, where it is often useful to introduce important or intuitive special cases before discussing the general case. It is to be hoped that an intelligent reader will understand that there is some connection between adjacent sentences in a paragraph. Sławomir Biały (talk) 14:37, 23 January 2011 (UTC)

I am not sure what you mean by "concentric style". Do you mean "concentrated style"? But this is precisely what I am objecting to. We are writing neither the encyclopedia britannica, nor a volume of Bourbaki. Rather, we are trying to ease the reader into learning about a subject he is presumably not familiar with. Cramming module coefficients into the lede even before explaining what they are is the best example I can think of of a latent bourbakism. Tkuvho (talk) 14:49, 23 January 2011 (UTC)

Please read the part of my post where I explained what I meant. I have already repeated this several times. Sławomir Biały (talk) 14:53, 23 January 2011 (UTC)

What you have explained is that the reader is apparently supposed to understand what the space of sections of the cotangent bundle of a manifold is in order to read a section of your lede. I think requiring the reader to know either what a manifold is or what a bundle is or what a section is amounts to an absurdity at the level of this page. Tkuvho (talk) 15:08, 23 January 2011 (UTC)

This is just silly. The lead doesn't even mention cotangent bundles. I was correcting your claim that there was somehow a mathematical error. I'm pretty sure there isn't a conceptual one. Sławomir Biały (talk) 15:22, 23 January 2011 (UTC)

Above you explained that you want to apply the universal construction of exterior algebra to the module consisting of sections of the cotangent bundle of the manifold. In order to understand the comment as it currently appears in the lede, the reader would have to be familiar with these notions. Notice that the using the module of smooth functions on the manifold as the "scalars" can safely be characterized as an advanced idea. Who are we trying to kid? That is, whom are we trying to teach, and whom are we trying to impress with the broadness of our knowledge? Tkuvho (talk) 15:38, 23 January 2011 (UTC)

Vector fields are a standard generalization of vector that we teach to first year students in the sciences. The scalars are understood as scalar fields. This is not an advanced concept. It requires no understanding of bundles. Sławomir Biały (talk) 15:52, 23 January 2011 (UTC)

Let's see if we can sort this out. You want to consider the functions as "scalars". You also want to consider the 1-forms as a module over such "scalars". If you want literally the same construction of the exterior algebra to apply in this setting, the 1-forms have to be an n-dimensional vector space over the scalars. Now a finite-dimensional vector space, as we know, has a basis consisting of n elements. Thus we need a basis of n elements, i.e. n 1-forms. This would imply that an arbitrary 1-form is a linear combination, with coefficients in the "scalars", of the basis 1-forms. Then indeed we can try to apply literally the same "universal" exterior algebra construction to this vector space. But there are at least two problems here. First of all, the "scalars" are not a field. Indeed, they contain divisors of zero. Far more seriously, one cannot have such a basis of 1-forms except in the case when the manifold is parallelizable. This would rule out all even-dimensional spheres, for example.

To be sure, this is not to say that such a construction does not exist. But it is not as simple as it appears from the lead. The amount of time it takes to sort this out seems beyond the ability of an average college junior. Tkuvho (talk) 17:24, 23 January 2011 (UTC)

It's not a free module - so what? — Kallikanzarid^talk 17:32, 23 January 2011 (UTC)

The symplectic form

It's currently mentioned right after the sentence "The exterior algebra contains objects that are not just k-blades, but sums of k-blades", which is misleading, in my opinion. I suggest to move it some place else, probably to the first paragraph. BTW, is symplectic form something a layman is usually intimately familiar about? ;) — Kallikanzarid^talk 14:52, 1 February 2011 (UTC)

Well, Bialy asked a few days ago about important applications of exterior algebra, and I think this is one of them. The symplectic 2-form is an element of the exterior square which is not a blade but rather a sum of blades. Note that this should not be confused with the closed differential 2-form which is also called the symplectic form, which is obviously a more advanced concept. Our page symplectic form concentrates on the linear-algebraic notion, I think. Tkuvho (talk) 14:56, 1 February 2011 (UTC)

The symplectic form illustrates something useful about higher-rank elements of the exterior algebra, so I have switched it with the sentence introducing the concept of rank. Sławomir Biały (talk) 15:19, 1 February 2011 (UTC)

Question for Kallikanzarid, is this misleading because it appears to suggest that the exterior powers consist of simple elements, or because it omits the fact that some elements are not homogeneous? Or some other reason? Sorry, I'm confused. Sławomir Biały (talk) 15:25, 1 February 2011 (UTC)

I'm not aware of anything that's called symplectic form other than a closed non-degenerate 2-form, and that one is a simple element. I'm confused :( — Kallikanzarid^talk 00:08, 2 February 2011 (UTC)

It's not simple, it's the opposite: it has full rank. I'm not sure where that leaves the discussion though. Sławomir Biały (talk) 00:27, 2 February 2011 (UTC)

I'm probably confusing terms. the symplectic form on M²ⁿ is a 2-form ω(X,Y) such that dω = 0 and Λⁿω is a volume form, right? A simple element is a k-form, right? — Kallikanzarid^talk 00:53, 2 February 2011 (UTC)

The term "simple" might be used in the sense of an element of the exterior algebra of "pure" order, in other words, living in a k-fold exterior power. I am more familiar with its use interchangeably with "decomposable", meaning that it is a wedge product of 1-forms. The differential structure is the next stage, when we look at the bundle of exterior algebras and their sections, when it becomes possible to apply the exterior derivative "d". Tkuvho (talk) 05:38, 2 February 2011 (UTC)

Ok, I don't mind this usage of the term. It would be great, though, to also give an example of a non-pure element. So far I haven't encountered one in differential-geometric applications. Also, "d" may be introduced for an abstract Lie algebra. It's still implicitly about corresponding (local) Lie group, but still :) — Kallikanzarid^talk 19:03, 2 February 2011 (UTC)

In differential-geometric applications, an important role is played by the identification of the exterior algebra with the Clifford algebra. Here the product in the Clifford algebra is captured as the sum of an exterior product and an interior product. This is important in geometric applications of spin bundles, index theorems, scalar curvature, etc. The best source for this is probably the book by Lawson and Michelson, "spin geometry". Tkuvho (talk) 20:10, 2 February 2011 (UTC)

WP:RANDY

Could folks who admittedly don't understand the topic and are unwilling to read reliable sources about it please post suggestions here instead of introducing potential errors or extraneous information into the lead of the article. Thanks, Sławomir Biały (talk) 14:24, 27 July 2011 (UTC)

/lambda^k notation

In the back of Spivak - Calculus on Manifolds it suggests in the addenda that the Lambda^k(V) notation for that vector space clashes with the notation for a quotient of the tensor algebra of V, so though the vector space in question is isomorphic to Lambda^k(V*) for finite V, the notation UpperOmega^k(V) was on its way to becoming standard. Should that change perhaps be made here?

Spivak - Calculus on Manifolds, p. 146 Addenda #4 (Jan 1998 printing) — Preceding unsigned comment added by 72.188.29.152 (talk) 22:27, 22 September 2011 (UTC)

Skullfission (talk) 22:30, 22 September 2011 (UTC)

That seems reasonable to me. Sławomir Biały (talk) 23:34, 22 September 2011 (UTC)

Confusing sentence

The following (partial) sentence from the current article is sufficiently, almost humorously, ambiguous to the extent that, although I think I know what it means, I can't be sure.

The k-vectors have degree k, meaning that they are sums of products of k vectors

Since I'm not sure I understand this, I can't actually fix it in the article, but here is my best guess:

A k-vector that is the sum of products of n ordinary vectors is said to have degree n

I hope someone who does know, will fix it. Dratman (talk) 04:36, 17 December 2011 (UTC)

It makes sense to me. E.g. for k = 2 you could say

all 2-vectors are sums of pairs of vectors.

It's counter-intuitive as in three dimensions all 2-vectors are 2-blades: you don't need to do any sums as every 2-vector is a product of two vectors. But for higher dimensions > 3 you do need to use sums for some ranks. It's maybe a little redundant as it's a restatement of e.g. this sentence from earlier:

The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector.

But it then leads into another point so can't jut be removed. The whole introduction could do with copy-editing for length which would probably deal with it.--JohnBlackburne^words_deeds 06:04, 17 December 2011 (UTC)

@Dratman: A "vector" is reserved to mean a 1-vector, or ordinary vector as you call it. This is not made adequately clear in the lead.

@JohnBlackburne: I'm sure you meant "all 2-vectors are sums of products of pairs of vectors". Which means, of course, any k-vector may be written ∑_i∏^k
_j=1v_i,j, where the v_i,j are "ordinary" vectors. A copyedit is badly needed; the lead is quite difficult to read. — Quondum^t_c 09:13, 17 December 2011 (UTC)

Yes, oops. Written quickly and trying to write it more accessibly. And yes, the lead is far too long with three much too long paragraphs, so needs needs probably halving in length and a complete rewrite for anyone with the time.--JohnBlackburne^words_deeds 09:19, 17 December 2011 (UTC)

There is absolutely no consensus for any kind of extensive rewrite of the lead. It is not "far too long": rather it is well within the established norms for article leads. It might be a little unusually long for a mathematics article, but most of our leads are too short by wider Wikipedia standards. As it currently stands, it explains very clearly the subject of the article in terms that are aimed at a layman. Each point it makes is dealt with in detail later in the article: the purpose of the lead is to provide a capsule version of the article that can (as far as reasonably possible) be understood by all likely readers. The lead has been hailed as an exceptionally clear explanation of a topic that is, by its very nature, hard for lay persons to understand. I'm not saying that copy editing isn't needed, but I disagree most emphatically with your call for a "complete rewrite". Sławomir Biały (talk) 12:32, 17 December 2011 (UTC)

(I should add that an enormous amount of effort was spent bringing the lead into its present form, including many edit-hours of work, peer review, and discussions both on this discussion page and at WT:WPM.) Sławomir Biały (talk) 13:59, 17 December 2011 (UTC)

Sorry for not replying sooner, but I definitely need to clarify. The main guideline on lead length is they should be no more than four paragraphs, and this certainly complies with that. But the assumption is that paragraphs are of reasonable length. The ones here are easily two or three times too big, and would normally be split for readability by anyone editing the article. But that would generate six or more paragraphs which is too many. The rest of the article is much better, as is especially clear from the Applications or History sections. The latter, with four reasonable sized paragraphs, is much closer to the ideal size and style of a lead section.

The last discussion on it seems to be this one, which I had forgotten about. I note there that I and another editor agreed that it was too long, but little seems to have been done, or at least it is still far too long. A third editor in this thread thinking it needs copyediting for readability, the main cause of which again is the overlong paragraphs.--JohnBlackburne^words_deeds 11:46, 1 January 2012 (UTC)

I agree with Sławomir Biały that the lede should be left alone as per WP:IAR. Tkuvho (talk) 12:07, 1 January 2012 (UTC)

Personally, I think the paragraphs are of reasonable length. Again, I have noticed a trend in mathematics editors that seems to favor extremely short paragraphs. For instance, compare this article with most of the paragraphs at Peloponnesian war. But regardless, if you have specific suggestions, I am willing to work with you. Complaining about the size of the lead, and saying that the whole thing needs to go just because of the length does not seem to me to be constructive. (Indeed, you show no indication of even having read the lead.) After the appearance of this thread, I did trim some of the fat and tried to improve the overall flow. But I don't really see anything substantial that can be cut altogether while keeping within the recommendations of both WP:LEAD and WP:MTAA. The only thing that really can be done would be to tighten up the wording by eliminating some of the explanations—such as saying what associativity means, what a graded algebra is, etc.—but this would almost certainly make the lead less accessible to lay persons. Sławomir Biały (talk) 12:54, 1 January 2012 (UTC)

I did not write that 'the whole thing needs to go', but given it's length a lot would need removing, and what's left reworking, to make it of reasonable length. That would involve removing some explanations but that is normal: provide a wikilink so readers unsure what e.g. associativity means can find out (and such probably should not have started with this article), those who know what it means can read on in confidence. There's no need to explain so much in the lead, especially terms and concepts that readers should understand before tackling the article.

A good example that comes to mind is the maths featured article 0.999.... The paragraphs in the lead are about the right length: some shorter, some longer but none excessively so. Or 1 − 2 + 3 − 4 + · · ·. Neither has "extremely short" paragraphs in the lead but reasonable length ones. Or the history section of this article. Many paragraphs are are too long at Peloponnesian War but it's C-class article, the lowest standard for a substantial article, and needs a lot done to improve it. The best guidance of this I've found is here; well outside of mainstream guidance but the more general guidance in the MOS, featured and good article criteria to write clear, consistent prose.--JohnBlackburne^words_deeds 13:43, 1 January 2012 (UTC)

The current lede of this page is already written in clear, consistent prose, due to the efforts of contributing editors. The lede of 0.999... was recently considerably trimmed down, to the detriment of the article in my opinion. Tkuvho (talk) 13:53, 1 January 2012 (UTC)

John, you wrote that the lead needs a "complete rewrite". These were your exact words. I'm saying that the lead of this article is just fine the way it is. It is clear and concise, and everything is clearly explained in language suitable for a general reader with a little mathematical background (such as an engineer). The paragraphs are longer than those at 0.999... because the notion of the exterior algebra is much more abstract and requires more explanation for a lay audience. Indeed, I can summarize that article in just a few characters: "0.999... = 1". From this minimalist perspective, the lead there is much too long! By contrast, this article needs to describe the exterior product and the exterior algebra. These are both very abstract things (to a nonmathematician) requiring a lot more text to convey the main ideas. This article has had lots of problems in the past with some readers not having the background to understand what an "associative algebra" is (nor even what the word "algebra" means). Over the years, many have complained about this here, and some at WT:WPM. The idea that a web of links should be sufficient to understand an article is popular among mathematics editors, but I'm saying that this didn't work here and the lead had to be rewritten. We aren't going to go back to how it was, especially not after I and other very experienced editors have spent many many hours bringing the lead to its present form. Sławomir Biały (talk) 13:57, 1 January 2012 (UTC)

Hear, hear! Tkuvho (talk) 13:59, 1 January 2012 (UTC)

"complete rewrite" was too terse and I've expanded and clarified it since. I don't accept your more general argument: by it the more advanced a mathematical article is the longer it's lead needs to be, to include all the prerequisites and definitions needed. But this goes not only against WP:LEAD but I can't think of another mathematics article with as long or longer leads, even though there are many more advanced. Most are much, much shorter.

I would appreciate a link to the WT:WPM discussion, a search does not turn it up. I'm not proposing it goes back to any previous version (I've not even looked at one), but improvements have to be based on it's state now, not how much work was done getting it to this state. The encyclopaedia is never done and certainly this article is far from featured article or otherwise unimprovable.--JohnBlackburne^words_deeds 14:38, 1 January 2012 (UTC)

Here is the discussion at WT:WPM. I don't accept your corollary to my general argument that "the more advanced a mathematical article is the longer it's lead needs to be." The requirements of WP:MTAA are that the lead needs to be as accessible as possible to the widest likely audience. This discussion page shows that the audience for this article includes students, engineers, physicists, medical scientists (!), and other non-mathematicians. That's simply an empirical fact. The lead needs to be accessible to such a group, much more so than the lead to an article like Cartan connection or automorphic form would need to be. Sławomir Biały (talk) 15:18, 1 January 2012 (UTC)

Also, I'm not saying that the article is perfect as it is now. But I don't see arguments based solely on length as constructive. The lead of the current article is a little long, yes. What I have been trying to convey is that the lead, length included, is this long for good reasons, and a lot of discussion and work has been put into it. If you have any other suggestions that might help to improve this article, then please make them here. If you see some explanation that can be made in fewer words without sacrificing its understandability, by all means implement it. But calling for a "complete rewrite" is simply unacceptable in view of the many many discussions that have taken place. In fact, in your very own words: "improvements have to be based on it's [sic] state now." Sławomir Biały (talk) 15:41, 1 January 2012 (UTC)

Recent edit by User:PsiEpsilon

Referring to this edit, aside from the fact that it seems hardly "motivating" to a casual reader to talk about areas in C² as opposed to the Euclidean plane that we learned about when we were small children, the entire section becomes dubious over the field of complex numbers. Indeed, the factor $ad-bc$ is then complex, and it can no longer be considered a "signed area". In fact, the wedge product over the complex numbers is not related to area. For instance, in C¹, we have $1\wedge i=0$ despite the fact that the parallelogram with sides 1 and i has area 1. Sławomir Biały (talk) 13:31, 26 December 2011 (UTC)

That's because 1 and i are scalars. They're all point like. For example, $1\wedge 2=0$ although the area of the rectangle between them is 2. The article makes it seem that ${\mbox{Span}}\left({\hat {\imath }},{\hat {\jmath }}\right)=\mathbb {R} ^{2}$ which is false. It should be ${\mbox{Span}}\left({\hat {\imath }},{\hat {\jmath }}\right)=\mathbb {C} ^{2}$.

If this comment of mine is not accepted, I guess we have to lose hope on wikipedia being unbiased. — Preceding unsigned comment added by PsiEpsilon (talk • contribs) 11:49, 30 December 2011 (UTC)

1 and 2 span a line segment, which has zero area. 1 and i are vectors in the complex plane, and are the sides of a rectangle there. (Draw a picture. This is very elementary.) Finally, the real span of i and j is R², not C². In any event, as I've already pointed out, it is false that the area of a parallelogram in C² is measured by the exterior product (at least, for any interpretation of the word "area" that seems reasonable to me). You are invited to try to make rigorous sense of it for yourself. Sławomir Biały (talk) 12:02, 30 December 2011 (UTC)

The interpretation of a complex number spanning a plane in the exterior algebra does not apply (as stated, they are simply scalars w.r.t. the algebra), so the "dubious" bit is not really an argument (application can be found alongside a complex Clifford algebra). On the other hand, if the exterior algebra over complex numbers (or any other field or ring) is to be presented, it only makes sense to do so if this is done with a suitable citation. Where it was introduced did not work well: under Exterior algebra#Motivating examples; it confuses rather than motivates. Complexifying the real case should be done later and more clearly if it is to be done at all. — Quondum^t_c 19:49, 30 December 2011 (UTC)

C¹ is a one-dimensional complex vector space over itself. It is indeed germane that the exterior product of two vectors in this space does not measure the area of the parallelogram defined by those two vectors. But if you prefer a two complex-dimensional example, let e₁, e₂ be a basis of C². The two vectors e₁ and i e₁ are the sides of a square. What is the area of that square? What is the wedge product of these two vectors in $\wedge ^{2}\mathbf {C} ^{2}$? Do you still believe that there is some connection between areas and the exterior product over C? Sławomir Biały (talk) 20:33, 30 December 2011 (UTC)

(Discussing areas in the complex case requires introducing the complex conjugate vector space, introducing new basis vectors ${\overline {e}}_{1},{\overline {e}}_{2}$ that are linearly independent from e₁ and e₂. This is all well-understood, but the section becomes completely wrong for the reasons I have stated if R² is naively replaced by C².) Sławomir Biały (talk) 20:49, 30 December 2011 (UTC)

1,2,i,2i, 3+700i etc are all scalars so they're point-like. Thus the exterior products are all 0. Clarifying further, for some scalar c, $x\wedge cx=0$. Complex numbers ARE scalars. It says so on the page http://en.wikipedia.org/wiki/Vector_space.

The function "Span" means "Complete Span" or,...

$\operatorname {Span} \left({{\mathbf {v} }_{1}},{{\mathbf {v} }_{2}},...,{{\mathbf {v} }_{n}}\right)=\left\{\left.\sum \limits _{k=1}^{n}{{{c}_{k}}{{\mathbf {v} }_{k}}}\right|\left.\forall k\right|1\leq k\leq n;{{c}_{k}}\in \mathbb {C} \right\}$

Some unconventional notations:

$\operatorname {R} \operatorname {Span} \left({{\mathbf {v} }_{1}},{{\mathbf {v} }_{2}},...,{{\mathbf {v} }_{n}}\right)=\left\{\left.\sum \limits _{k=1}^{n}{{{c}_{k}}{{\mathbf {v} }_{k}}}\right|\left.\forall k\right|1\leq k\leq n;{{c}_{k}}\in \mathbb {R} \right\}$

$\operatorname {I} \operatorname {Span} \left({{\mathbf {v} }_{1}},{{\mathbf {v} }_{2}},...,{{\mathbf {v} }_{n}}\right)=\left\{\left.\sum \limits _{k=1}^{n}{{{c}_{k}}{{\mathbf {v} }_{k}}}\right|\left.\forall k\right|1\leq k\leq n;{{c}_{k}}\in \mathbb {I} \right\}$

I've already said, C is a one-dimensional complex vector space over itself, so it is indeed meaningful to talk about the wedge product (it's identically zero). It's also two-dimensional in the geometrical sense, so it's also meaningful to talk about area. I'm saying that these two notions do not coincide. The same holds in higher dimensions as well (I've already given an example of this, though it should be obvious to anyone with some mathematical maturity). Finally, you are confused about the notion of a vector space. There is no presumption that the field should be the complex numbers (or algebraically closed, etc.) The ground field is part of the data involved in specifying a vector space. For instance, while C is a one-dimensional vector space over itself, it is a two-dimensional vector space over R. As I've already indicated, the section under discussion (Exterior algebra#Motivating examples) is wrong if the ground field is C. It depends essentially on the ground field being R. If you can't understand the arguments I have made, just take my word for it. Sławomir Biały (talk) 13:28, 1 January 2012 (UTC)

You seem to keep wanting to get get the area of a scalar. Forming any algebra over a field or ring, AFAICT as a nonmathematician, seems to be a process independent of the field or ring over which it is. So you have to deal with the scalar abstractly, be they real numbers, complex numbers, a finite field or even quaternions. You refer to the basis of C² being e₁ and e₂, and then promptly start talking about e₁ and i e₁ as though they are linearly independent and describing a planar geometric figure, whereas neither is true. e₁ ∧ i e₁ = 0, whereas e₁ ∧ i e₂ = i(e₁∧e₂) and has an "area" of i. The exterior product of two scalars, contrary to what you say, is not zero but equal to the product of the scalars. Do not get confused by the vector space over which the algebra is formed (which in this example is spanned by {e₁, e₂}), and the algebra itself, which is also a vector space (spanned by {1, e₁, e₂, e₁∧e₂}). — Quondum^t_c 18:20, 1 January 2012 (UTC)

The exterior product of two vectors in C¹ is zero, although it's true what you say that the exterior product of scalars within the algebra is their product. This is not what I mean, as should have been clear. I'm talking about the wedge product of elements of a one complex dimensional vector space C¹ (see my original post). This is identically zero (the exterior algebra is generated in degrees zero and one). On to my two-dimensional example: are you seriously asserting that it is not meaningful to talk about the parallelogram with sides $e_{1}$ and $i\,e_{1}$? I can define it formally as the convex hull of the points $0,e_{1},i\,e_{1},e_{1}+i\,e_{1}$. The area of this convex hull is equal to one, yet the wedge product of $e_{1},i\,e_{1}$ is zero. Again, draw a picture. This is elementary school level mathematics! Sławomir Biały (talk) 22:26, 1 January 2012 (UTC)

By the way, the article itself is neutral to the field. It is just the Motivation section where the field needs to be real. I agree absolutely that the exterior product makes sense regardless of the ground field. I am simply saying that it is not related to "area", so the motivation section becomes wrong if R is replaced by C. Sławomir Biały (talk) 22:33, 1 January 2012 (UTC)

In your original post you seem to have written 1 ∧ i = 0, with which I do not agree, but that is not of importance. The parallelogram to which you refer may be interpreted geometrically, though this is divergent interpretation from that in the exterior algebra, and should not be considered in the context of an exterior algebra. I agree that it is inappropriate to extend the concept of area to a complex vector space other than as an abstract analogy (as indicated by my putting the word "area" into quotes), and I agreed that including C² at that point did not work. As an example of the abtract use of words, an elliptic curve over a finite field is hardly a curve, but that is the name in use. I think that debating the detail is peripheral – no-one is suggesting that the C² should not have been removed. — Quondum^t_c 18:59, 2 January 2012 (UTC)

I wrote "in C¹". If I had meant that I was working with scalars, I would surely not have included a superscript. I have emphasized this in every post. Sławomir Biały (talk) 20:04, 2 January 2012 (UTC)

Well, let's just say I find your notation and wording confusing. "In C¹" could mean a number of things, for none of which I would have denoted a specific 1-vector as 1, since this is the notation for a scalar (specifically the identity element) in the algebra. At this stage I am disinclined to try and work out whether you meant the exterior algebra vector space-isomorphic to C¹ or the exterior algebra over the vector space C¹ (which has twice the number of dimensions). — Quondum^t_c 05:08, 3 January 2012 (UTC)

My point is that, over the complex, the exterior algebra has nothing to do with area (as it is understood to be a real number). I invite you to work out your own counterexample in whatever notation or setting you feel comfortable with, rather than quibble over the way I have chosen to explain it. Consider this an exercise. Sławomir Biały (talk) 11:29, 22 January 2012 (UTC)

I agree that we seem to be quibbling pointlessly, so let's keep the focus on the article. Is there something that you are suggesting still needs changing? — Quondum^☏✎ 12:06, 22 January 2012 (UTC)

No, I and another editors have already reverted the original edit that PsiEpsilon was edit-warring to include. But it would be nice to get solid consensus that this edit is definitely not acceptable. Quibbling with me over my choice of notation is not constructive. Sławomir Biały (talk) 12:21, 22 January 2012 (UTC)

I apologize for focussing in an unconstructive direction. And I agree that the edits incorporating C² made no sense, and must not be in the article. It would seem that Nageh through two reverts agrees as well. So perhaps you have your consensus. I noticed no further editors being involved. There was a short sequence of own reverts by PsiEpsilon, so to label it as edit-warring may be a bit harsh. — Quondum^☏✎ 13:35, 22 January 2012 (UTC)

Pity wikipedia is such a bias source of information... This is such a good article, except its bias. No hope in this page. — Preceding unsigned comment added by PsiEpsilon (talk • contribs)

It's a pity your edits to the article were complete nonsense. Thanks for playing, Sławomir Biały (talk) 13:13, 8 May 2012 (UTC)

Pity wikipedia that it is going to be destroyed by being bias against complex numbers.RefininementDrawsNear (talk) 13:50, 8 May 2012 (UTC)

No, there is no bias against complex numbers. The issue here is that the exterior algebra has nothing to do with area when regarded over the complex field. The section under discussion, whose purpose is to motivate the exterior algebra (not define it) says that the exterior algebra of two vectors measures the area of the parallelogram they span. This is actually wrong over the complex field. So, if there is a bias, it is against having wrong/misleading/meaningless information in our articles. Would you prefer to have a wrong article? Sławomir Biały (talk) 14:02, 8 May 2012 (UTC)

Let me clarify. It depends on whether you are going to be taking the Exterior Algebra in the "Clifford Algebra way" or the "Geometric Algebra" way. I know that sounds really non-rigorous, but...PsiEpsilon (talk) 00:43, 9 May 2012 (UTC)

No, actually it doesn't. The meaning of "area of a parallelogram" is unambiguous and has nothing to do with geometric algebra. Sławomir Biały (talk) 00:57, 9 May 2012 (UTC)

I'm talking about the fact that $\mathbb {C} ^{2}$ is still equipped by i-hat and j-hat.PsiEpsilon (talk) 01:04, 9 May 2012 (UTC)

Your comment is a bit obscure. I have taught linear algebra both over R and over C and I still have difficulty understanding your comment. Tkuvho (talk) 08:53, 9 May 2012 (UTC)

Seeing that the section under discussion is prominently labeled Areas in the plane, it's not at all clear that being able to define the exterior algebra for vector spaces over C is relevant (any more than defining it over, say, a finite field would be). Certainly, one can define the exterior algebra over any field, including the complex numbers, and indeed the article does so in the section labeled Formal definitions and algebraic properties. The issue here is that if the vector space is complex, rather than real, there is no longer any connection to the example of area, which the section under discussion is actually about. Sławomir Biały (talk) 13:02, 9 May 2012 (UTC)

Bad Organization

This article is badly organized, in the sense that it starts with the definition of a different term (wedge product), and only gets around to defining its own term in the second paragraph. And the first paragraph is a mishmash of interpretive statements that don't really define the wedge product. I think it would be better to define the wedge product in a separate article. Does anyone agree? Eleuther (talk) 11:05, 1 January 2012 (UTC)

Actually, exterior product redirects here, so it is appropriate for it to discuss the exterior product in the first paragraph. There is also no nontechnical formal definition, so it would be inappropriate to define it in the lead of the article. In fact, defining the wedge product actually requires defining the full exterior algebra (as a quotient of the tensor algebra, or by a universal construction). But if we were to define it this way in the lead, we would lose 99% of prospective readers (see the history of this discussion page for evidence). Sławomir Biały (talk) 12:41, 1 January 2012 (UTC)

What does "imitate" mean mathematically?!?

"This imitates the usual definition of the cross product of vectors in three dimensions." This is really imprecise. What does it mean?!? Is it the same thing? If not, why not (it looks the same!) — Preceding unsigned comment added by 193.206.68.168 (talk) 14:02, 20 April 2012 (UTC)

It is not the same since the wedge product in in a different vector space than the cross product. We don't have $e_{1}\wedge e_{2}=e_{3}$ for instance. Sławomir Biały (talk) 16:00, 20 April 2012 (UTC)

A request for clarification

In the early part of this article, with the example headed: Cross and Triple products, at the point where there occurs the text

"Bringing in a third vector...."

when looking at the form of the product of three vectors

$\mathbf {u} \wedge \mathbf {v} \wedge \mathbf {w}$

many readers will surely be very tempted to draw the conclusion that a wedge product of three vectors is equated with the result of performing the binary wedge operation twice in succession. In other words, that u ∧ v ∧ w may be identified with (u ∧ v) ∧ w and in particular that

$(\mathbf {e} _{1}\wedge \mathbf {e} _{2})\wedge \mathbf {e} _{3}=(\mathbf {e} _{3}\wedge \mathbf {e} _{1})\wedge \mathbf {e} _{2}=(\mathbf {e} _{2}\wedge \mathbf {e} _{3})\wedge \mathbf {e} _{1}=\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}$

But from what I understand looking down further to the remainder of the article, I suspect such a supposition would be incorrect. Might a sentence or two be inserted at this early point, within the example, to clarify this situation one way of the other?

This relates to a more general point that it might perhaps be usefully clarified very briefly, even within these early examples, what kind of object the result of a wedge product is. For example, that the wedge product of two 3-vectors is not itself (as most readers will already know that a cross product, by contrast, is) another 3-vector.

Thanks. — Preceding unsigned comment added by 83.217.170.175 (talk • contribs)

In answer to your question, the wedge product is associative, so u ∧ v ∧ w = (u ∧ v) ∧ w. But I think for the purposes of this elementary discussion it is more helpful to think of the wedge product of three vectors on its own, rather than as an iterated wedge product. For your second question, when you multiply elements of different degrees, the degrees add like polynomials. This is already mentioned in the lead. Sławomir Biały (talk) 13:18, 8 May 2012 (UTC)

Sławomir, thank you very much for your fast response. I would like to clarify briefly what is - and was when I asked the question - in my mind. Firstly, to me it does not follow from associativity that u ∧ v ∧ w = (u ∧ v) ∧ w. I understand associativity as a property of binary operations (in the sense of mappings from AxA to A, for some set A). Associativity says that, for such an operation, (u ∧ v) ∧ w = u ∧ (v ∧ w), for all triples (u,v,w) in A³. But my question was rather: Would I be correct, in this case, to identify the presented ternary operation, written u ∧ v ∧ w, and with the value as supplied in the text, as being equivalent to an iterated binary operation at all; or, instead in this case, is the ternary version of the ∧ operation rather to be interpreted as something separately defined in its own right, and not constructible out of the building block of the binary version? I did not find the answer obvious from the existing text of this example - And I in fact guessed (wrongly it now seems, for your reply - although I am still a bit unclear) that the answer was the latter: i.e. I guessed that u ∧ v ∧ w was not to be interpreted as an entity that I should assume could be built as a combination of binary wedge applications. (Of course, I understand that any associative binary operation in a simple way unambiguously defines a meaning for arbitrarily long finite expressions: a ternary, 4-ary, etc extension of itself. But this knowledge did not answer my question, since I was not sure whether for example e₁ ∧ e₂ even belonged to a set of objects to which the binary wedge operation could be applied.)

I think this illustrates a common difficulty of explaining mathematics to someone who is seeing it for the first time. Typically the reader (I in this case) will be asking themselves questions that are difficult for the writer to anticipate. (So for this reason, the reader will not always be finding the discussion as elementary as was intended.)

An important question is whether many of the readers who are not familiar with wedge product will - on reading this text - be puzzling in a way similar to the way that I was. I am afraid I don't know the answer to that. (I am a reader with probably a few thousand hours of previous exposure to different areas of abstract algebra; and also - some decades ago - a 1st class maths degree.)

Thanks again for your response. — Preceding unsigned comment added by 83.217.170.175 (talk • contribs) 17:51:59,‎ 8 May 2012 (UTC)

Note that in this context, when we refer to a k-vector in this context, the k refers to the grade of the entity, not the dimensionality. Thus 0-vector is a scalar (1-dimensional), a 1-vector is a standard n-dimensional vector (where n is the dimensionality of the geometric space), a 2-vector is what we call a bivector, and so on. I note you referred to a 3-vector, meaning what we'd call a 1-vector here.

The wedge product is a true binary operation, with true associativity in the ordinary sense, and the algebra as a whole is closed under the wedge product. What may be confusing is that the wedge product maps specific grades (distinct subspaces of the the algebra, sharing only the zero element) in a way that is not closed. In particular, it maps p-vector and a q-vector onto a p+q-vector. Thus, a 1-vector and a 1-vector are mapped onto a 2-vector (bivector). A 1-vector and a 2-vector (and also a 2-vector and 1-vector, as per the associativity) are mapped onto a 3-vector.

This all should be reasonably clear from the lead, but it is easy to have some misconception due to quirky terminology, such as your interpretation of a "3-vector". (We'd say the cross product of two 1-vectors in an algebra over a three-dimensional vector space is another 1-vector. The algebra is a larger space than the original vectors space. Try re-reading it with this in mind.) — Quondum^☏ 19:39, 8 May 2012 (UTC)

Excess implied algebraic structure in the lead

I am parking this as a note for a necessary correction to the lead (anyone feel free, I may get around to it in due course). The statement in the lead

The magnitude of u ∧ v can be interpreted as the area of the parallelogram...

suggests that the magnitude of a bivector is defined as part of the definition of an exterior algebra. This implication is incorrect: the definition of an exterior algebra (with the minimal structure that qualifies it as an exterior algebra) is without any concept of magnitude or interior product, though this doesn't prevent such structure from being added. It might even make sense to point out in the lead that no concept of magnitude is needed in the definition. Penrose, for one, stresses this in The Road to Reality. — Quondum 06:53, 6 September 2012 (UTC)

This very issue is discussed in a footnote. The point of saying this at all, though, is so that lay persons get an idea of what the wedge product is. Sławomir Biały (talk) 12:37, 6 September 2012 (UTC)

Good point about giving an idea. It was slack of me not to read the footnote, though it'd be good to make this more blatant. What is always true is that the scalar multiplier of a parallel reference bivector (without using the structure of an interior product) will be the same as for the area of the parallelograms with any interior product whatsoever (except for null vectors, when the areas will be zero). It would be nice to find concise wording that can draw on this proportionality concept without resorting to the structure of an interior product. — Quondum 16:01, 6 September 2012 (UTC)

This probably takes too many words I think to do conpellingly in the lead. The proportionality idea is discussed somewhat in the motivation section. Sławomir Biały (talk) 12:30, 8 September 2012 (UTC)

Rank of a ... multivector?

The lead says

The rank of any element of the exterior algebra is defined to be the smallest number of simple elements of which it is a sum.

While I can make sense of this statement, it seems to have no value to me, whereas a similar statement ties up with the more normal and useful concept of rank:

The rank of any k-vector is defined to be the smallest number of simple k-vectors (i.e. k-blades) of which it is a sum.

Considering that I've just removed a whole lot of the curious term k-multivector, I wonder whether this might be part of a similar confusion. Can anyone enlighten me about the intended meaning here (or even just correct the lead)? — Quondum^☏ 14:02, 2 September 2012 (UTC)

Yes, this is what was meant. Feel free to implement the change if you think it's clearer. Sławomir Biały (talk) 00:04, 6 September 2012 (UTC)

Will do. What might not have been clear is that "any element of the exterior algebra" can be the sum of nonzero elements of differing grades, whereas "any k-vector" cannot be. This makes the two statements nontrivially different in meaning, and hence it is a matter of correctness, not merely a matter of clarity. — Quondum 06:28, 6 September 2012 (UTC)

Yes, right. It should have read "homogeneous element". Sławomir Biały (talk) 12:40, 6 September 2012 (UTC)

I'm not exactly sure what the issue is with the term "k-multivector". This term cerntainly appears in the literature outside of Wikipedia. I don't object to the change though. Sławomir Biały (talk) 13:06, 6 September 2012 (UTC)

I hadn't encountered it before this article, and the article used both terms apparently without highlighting their equivalence (though I do see that it was defined in the article). I've not encountered it in any other WP article, though I see it occurs in Multivector, which hapazardly switches between the terms without indicating their equivalence, or even defining a k-multivector. If its use is notable, then we should make this clear as an alternative term for k-vector. In some ways, using "k-multivector" (as in "a homogeneous grade-k multivector") is preferable, as it potentially removes the confusion with "k-dimensional vector". The term k-multivector (or its equivalents, p-multivector etc.) seems to occur in only a handful of books, whereas k-vector or p-vector seems to be about three orders of magnitude more prevalent. — Quondum 12:19, 8 September 2012 (UTC)

My thoughts as well, that "k-multivector" is marginally preferrable, because of the potential for confusion over the term "k-vector". But I don't want to insist on it. Sławomir Biały (talk) 12:28, 8 September 2012 (UTC)

It is in the lead where the potential for confusion is the greatest, so it probably makes sense to draw attention to not-to-be-confused-with terms such as 4-vector (I've had to help someone out of this confusion before, I forget where). In association with this, the alternative and unambiguous term k-multivector could be mentioned (in the lead or later), provided that we feel that it is adequately notable. — Quondum 12:57, 8 September 2012 (UTC)

I've dealt with this by adding a footnote in the lead. Since it is only a footnote, I felt the burden of notability to be lessened, and included the mention of a k-multivector.