Wikipedia:Featured article candidates/Vector space/archive1

The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was not promoted by SandyGeorgia 16:08, 25 January 2009 [1].

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Vector space

Nominator(s): Jakob.scholbach (talk)

I'd like to nominate this article for FAC. It has passed its good article nomination, has had a peer review, whose comments have since been included. I believe the article to meet the FA criteria. Principal contributors in terms of edit counts are myself and Silly rabbit. Thank you for the review.—Preceding unsigned comment added by Jakob.scholbach (talk • contribs) 15:52, January 17, 2009

• Oppose per criterion 1.(c). Just looking at the lead:
  • ... tuples of real numbers such as Euclidean vectors ... – and "Euclidean vector" is about physical vectors. The electric field vector at the point one metre above my centre of mass as of 00:00, 17 January 2008 (UTC) is not a triple of numbers per se. (It can be specified by a triple of numbers if you choose a basis). BTW, don't assume that everyone knows what the heck a tuple is. This is the second paragraph of the lead. See WP:NOT PAPERS, point 5 (and note that it is a policy, not a guideline).
  • ... they are completely specified by a single number called dimension. I think this is supposed to mean that all vector spaces with the same dimension are isomorphic to each other; but that's not how someone who doesn't already know this would understand this sentence. And, even with my interpretation, I think this only applies to finite-dimensional vector spaces.
    • FYI, it applies to all vector spaces. Algebraist 17:22, 17 January 2009 (UTC)
      • (I'm striking that out then. -- Army1987 – Deeds, not words. 17:31, 17 January 2009 (UTC) BTW, out of curiosity, how does one prove that? Does one need the axiom of choice? -- Army1987 – Deeds, not words. 17:33, 17 January 2009 (UTC))
        • Yes. The statement that every vector space has a dimension is ZF-equivalent to full AC. Algebraist 17:38, 17 January 2009 (UTC)
• I won't even bother to finish reading it. -- Army1987 – Deeds, not words. 17:13, 17 January 2009 (UTC)

Do I understand you right, Army1987, that you see a factual error in the lead section? By the guideline you cite yourself, the lead should be an accessible overview, detailed explanations on choices of bases should (and do) come in the body of the article. While Euclidean vectors are ad hoc a physical concept, they do convey the right understanding of (mathematical) vectors and vector spaces, and were indeed one of the historical roots of them, so I think it is appropriate to mention them at this point.

As for "they are completely specified by a single number called dimension." I fail to see what possible wrong intuition could be implied by that wording, but perhaps a layman reviewer could tell? Jakob.scholbach (talk) 18:12, 17 January 2009 (UTC)

@Euclidean vectors: I have reworded this a little. Notice, though, that the corresp. article says "In three dimensional Euclidean space (or R3), vectors are identified with triples of numbers corresponding to the Cartesian coordinates of the endpoint". Jakob.scholbach (talk) 18:18, 17 January 2009 (UTC)

Perhaps write, instead of "vector spaces are completely specified by a single number called their dimension" (which is hard to "visualize" by a non-mathematician), write "for each natural number n and each field K, all vector spaces over K having dimension n are isomorphic, i.e the "same". In particular, this implies that all vector spaces over a particular field of scalars, are completely specified by a single number called the dimension.". --PST 23:41, 17 January 2009 (UTC)

That wouldn't cover the infinite-dimensional case. Algebraist 01:19, 18 January 2009 (UTC)

I think the wording of the first sentence I cited implies that Euclidean vectors are a particular type of tuples of natural numbers, while the reverse is true. As for the "infinite-dimensional case", in the current version "number" does link to cardinal number, although anybody who doesn't follow that link won't consider infinity to be a number. (If a way to address these issues occurs to me, I'll point it out.) -- Army1987 – Deeds, not words. 02:50, 18 January 2009 (UTC)

OK. I have reworded it to avoid the implication made you are referring to. As for the possibly infinite dimension: I also think that lay readers won't think of infinity (or even a particular type of it, countable/uncountable) as a number. However, it would be less correct to write, say: "They are specified by a number called dimension, which may be infinite.", because this implies that all spaces of infinite dimension are isomorphic, which is not true. At the moment I don't see a good way of a) not telling what a cardinal number is (as opposed to the every-day meaning of number) and b) making a correct statement. We could just remove that part of the sentence. What do you think is best? Jakob.scholbach (talk) 11:37, 18 January 2009 (UTC)

For now I've just unhidden the word "cardinal" from that link, but it should still be made clearer. -- Army1987 – Deeds, not words. 11:43, 18 January 2009 (UTC)

I chose to remove that bit of information. As I saw when re-reading the "motivation and definition" section, the possibility of being infinite is mentioned, and the precise statement (i.e. mentioning "cardinality" is given below), so this bit of information may be omitted in the lead. Jakob.scholbach (talk) 14:43, 18 January 2009 (UTC)

Thanks for your recent change, which is better than what we had before. I trimmed it down a bit. I hope that settles this matter? Jakob.scholbach (talk) 15:37, 18 January 2009 (UTC)

Comments - sources look okay, links checked out with the link checker tool. Ealdgyth - Talk 15:58, 18 January 2009 (UTC)

• I've struck out the "Oppose" as the lead has been improved. If I have time to take a more careful look at the rest of the article, I'll decide whether to support. -- Army1987 – Deeds, not words. 16:58, 18 January 2009 (UTC)

• Comment - The section on vector bundles is good but has some flaws. For example, to explain why the Mobius line bundle is different from the cylinder globally (i.e why the bundle is non-trivial), the section refers to orientable surfaces. While this is OK if you are dealing with smooth bundles, it is not in this case. Orientability is a property of differentiable manifolds and since "looks like" in the image caption links to homeomorphism, you are not in actual fact explaining why this is a trivial bundle (homeomorphisms do not necessarily preserve orientability). In effect, orientability only distinguishes differentiable manifolds but not topological spaces that are also differentiable manifolds (and I think you are dealing with the category of topological spaces here). Some parts of the section have to be rewritten. Note that, it is indeed true that globally the Mobius strip is not homeomorphic to the cylinder but the current explanation is not correct. --PST 09:14, 19 January 2009 (UTC)
  • One can define orientability as a purely topological property of manifolds. Orientability#Topological definitions mentions some ways of doing this. Algebraist 10:23, 19 January 2009 (UTC)
    • Yes, but my point is that orientability is not preserved under homeomorphisms. Therefore, the explanation in "vector bundles" is not adequate. --PST 15:11, 19 January 2009 (UTC)

No. You might want to read the article Algebraist is referring to. It says (correctly) "A compact n-manifold M is orientable if and only if the top homology group, $H_{n}(M,\partial M;\mathbb {Z} )$, is isomorphic to $\mathbb {Z}$". A similar statement holds for non-compact manifolds, where on has to consider (co)homology with compact support. Homology groups are a purely topological invariant, i.e., given a homeomorphism M → N, the groups are isomorphic. Therefore the statement in the article is right. Jakob.scholbach (talk) 15:18, 19 January 2009 (UTC)

I myself work in homology theory (please don't attack (in the second last sentence you implied more than just "I don't know what a topological invariant is")). It is just some parts of differential topology which I am not familiar with. I might be confused but isn't it true that the map f(x, y) = (−x, y) is a homeomorphism of the plane, whereas it has negative Jacobian determinant and therefore is not orientation preserving? --PST 21:53, 19 January 2009 (UTC)

Sure, homeomorphisms need not preserve orientation, but they do preserve orientability. Algebraist 22:44, 19 January 2009 (UTC)

I was seriously confused! Yes, I wasn't thinking clearly (I knew that but was thinking something else). Thanks for pointing that out Algebraist. --PST 22:49, 19 January 2009 (UTC)

I've been watching this thread. In partial defense of PST, I would note that at Orientability#Topological definitions, two of the definitions are somewhat dubious here. The first says a surface is not orientable if there is an embedded Moebius band. The second says that transition functions preserve orientation. Both rather beg the question! However Jakob has centred the point that this is a topological invariant, even if it is hard to make that intuitively clear. Consequently that would clearly be the wrong way to explain the issue in this article. The current article doesn't attempt such an explanation and I am happy with the current text. Geometry guy 23:08, 20 January 2009 (UTC)

• Comment (I don't know the best way to express my opinion without sounding too critical. Please know that all I want to accomplish by this post is to give some feedback.) While the article is well developed (in fact, very well-developed), it feels very short in many important points or facts: categorical point of view (a sentence is nearly enough), 1st, 2nd (and 3rd?) isomorphism theorems, their roles in homological algebras, why historically algebraists preferred to study rings or fields instead of vector spaces, important examples of vector spaces over fields other than real or complex, rank-nullity theorem, algebras (as generalization of vector spaces), application to physics (mostly in forms of Hilbert spaces), roles of vector spaces in the study of division algebras skew-fields, meaning and significance of basis changes (for example we are often interested in how two vector spaces are isomorphic not just isomorphic) relationship between annihilators and orthogonal complement, applications of duality in linear algebra (e.g., min-max principle, not only ones in functional analysis), a viewpoint of a vector space as an abelian Lie algebra, geometrical issues such as convexity with connection to dimensionality. -- —Preceding unsigned comment added by TakuyaMurata (talk • contribs) 21:29, 19 January 2009 (UTC)

Many thanks for your comments and, BTW, absolutely no need to apologize!

I have two general responses to your points: First, I have to say what you are probably already aware of, that the article has a limited total length. With this in mind, and with the aim to give reasonable amount (i.e., sufficiently to be able to explain things briefly, not just mention them) of space to must-have topics, it is difficult to include everything that comes to mind. Second, esp. where applications are concerned: in some sense most mathematics, esp. most applied mathematics can more or less reasonably be formulated in terms of vector spaces. I think the article has to focus on applications etc. where mainly (or only) the vector space structure of some given entity is used or studied. Using this guideline, topics effectively talking about polynomial rings, say, should be treated very briefly here or even not at all.

• categorical point of view: The article says "The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field F) is an abelian category." What else do you think is important? I personally think the category of v.sp. is of comparatively little importance, compared to, say, the category of modules over some ring, (both for this article and for math in general) since it is so simple. One could think about mentioning semisimplicity, but I'm not sure that is a core piece of information.

• Is reworded and slightly expanded. I think that's as much as we can afford. Jakob.scholbach (talk) 23:09, 19 January 2009 (UTC)

• Isomorphism theorems: we do have the 1st one. I'll think of a rewording so as to allude to the other ones.

• Done. Jakob.scholbach (talk) 23:09, 19 January 2009 (UTC)

• Role of vsp in homological algebra: what do you have in mind, specifically?
• Historical interest in rings vs. vsp: I have to say I'm not historically versed enough to even make sense of that point. At least to the modern understanding, as it is, vector spaces are of a pretty much different flavor than rings or fields, right? Please expound a bit...
• Vsp over other fields than R or C. What do you think of?
• rank-nullity: well, we have the first isomorphism theorem, which is the same thing. I'll try to reword that part (also w.r.t the 2nd and 3rd iso thm.).

• Done. Jakob.scholbach (talk) 23:09, 19 January 2009 (UTC)

• algebras: there is a whole section on that. What do you miss specifically?
• applications to physics/Hilbert spaces: likewise (see the final paragraph in the Hilbert space section)
• basis change: OK. that's a bit short. I'll expand that a bit.

• First draft done (in the section on linear maps). I'll revisit later. Jakob.scholbach (talk) 23:09, 19 January 2009 (UTC)

• I've also added an illustration depicting different choices of bases. Jakob.scholbach (talk) 14:24, 24 January 2009 (UTC)

• division algebras: I think about how to add a sentence to the algebra section.

• I have added a statement about f.d. real division algebras (in the section on vector bundles, after K-theory). Jakob.scholbach (talk) 22:51, 20 January 2009 (UTC)

• annihilators & orthogonal complement: I think that does not belong here, I would put that to inner product or bilinear form.
• applications in linear algebra: that sounds reasonable

• I have added a word about the minimax theorem. Jakob.scholbach (talk) 16:54, 23 January 2009 (UTC)

• vector spaces as abelian Lie algebras: Do you mean "[x, y] := 0"? I don't think that's an important piece of information here. Where is that relevant? If anywhere, I'd put that to Lie algebra.
• convexity: perhaps that should go to topological vector space? Jakob.scholbach (talk) 22:19, 19 January 2009 (UTC)

Comment

I've made some comments about the reciproqual vector spaces in the peer review that haven't been adressed. You might want to look into that. I may or may not opine later, but not having a section on the peciproqual vector spaces is a somewhat major issue.Headbomb {^ταλκ_{κοντριβς} – WP Physics} 06:15, 20 January 2009 (UTC)

Somehow your comment in the PR went unnoticed by me. Sorry about that. I'll put a sentence about that into the Fourier section, perhaps alluding to the general principle behind the Fourier transform. (A section is clearly more than we can afford). Jakob.scholbach (talk) 10:42, 20 January 2009 (UTC)

I have added a sentence about that, together with its mathematical explanation, Pontryagin duality. All right? Jakob.scholbach (talk) 22:51, 20 January 2009 (UTC)

• Oppose This isn't a particularly esoteric mathematical concept; it ought to be possible to explain it to laymen in the lead much, much better than we do now. I realise that such leads are not uncommon in mathematical articles, but, per WP:LEAD, they have no right to be, and we ought to at least make an attempt to explain it in a way that, say, an A-level mathematics student, or first year university student could understand. Shoemaker's Holiday (talk) 06:57, 20 January 2009 (UTC)

• I share your wish to make the article esp. the lead as accessible as possible. The following reviewer gives some concrete points which I will address. Could you please point out other parts of the lead or elsewhere where you think accessibility is not as good as possible? Jakob.scholbach (talk) 10:42, 20 January 2009 (UTC)

• The first section of the lead is now better. I realize that the following ones do contain technical terms like "analytic geometry" or "matrices", but I think we can't avoid these totally. Jakob.scholbach (talk) 18:48, 20 January 2009 (UTC)

Let's look at the first illustration, as I think it shows one of the problems with the article. If you know what it's doing, it's clear. If you don't, you won't understand vector addition or scalar multiplication after looking at it. A much better way to illustrate it might be show v. Show w. Show v+w as w coming off of the endpoint of v. Show 2w as 2 seperate ws, one after another.

For these kind of esoteric articles, you really need someone with a decent background, but little knowledge of the specific concepts that can ask you questions and force you to clarify as much as possible. I don't think it's at quite that level yet. Shoemaker's Holiday (talk) 10:31, 24 January 2009 (UTC)

Well, the illustration does show two copies of w, and from looking at the image, it should be clear (without background, I hope) that the new arrow is twice as long as the old one (In addition I reworded "scaled" in the image caption to "stretched"). Likewise the image does show a (faint) copy of w which is appended at the end of v and the sum is the diagonal of the parallelogram constructed this way. I fail to see how that could be clearer. (You might object that the faint copies of w and v used for the parallelograms don't carry arrowheads, but this is simply because the arrowheads would overlap and it would be hard to distinguish what goes on). Additionally to that illustration, we (now) do have more illustrations (in the motivation section) that explain the matter once more (and giving the explanation of the constructions in words). (Also, there are subarticles on vector addition and scalar multiplication. They could (and should) be better, but this is a different story). I stand ready to be corrected, but I feel the reader has to have a certain willingness to think about what might be new to her/him, and to accept an "esoteric article" like this as an invitation à la danse (macabre? ;), not the dance lesson (including instructor) itself. Jakob.scholbach (talk)

14:24, 24 January 2009 (UTC)

P.S. In line with the above, I'd like to say that the objective of a lead section is not (and cannot be, IMO) to make the topic understandable to the last bit, but give an overview. More details are to come (and do come, as pointed out above) in the body. Jakob.scholbach (talk) 14:31, 24 January 2009 (UTC)

Please read WP:LEAD: "While consideration should be given to creating interest in reading more of the article, the lead nonetheless should not "tease" the reader by hinting at—but not explaining—important facts that will appear later in the article.", and WP:NOT PAPERS: "A Wikipedia article should not be presented on the assumption that the reader is well versed in the topic's field." Seriously. We're not talking about quantum gravity. It is completely possible to explain what a vector space is in a way that the average junior high school student could understand (provided he cared about it), contains no lies, and is no longer than four paragraphs. Probably such a junior high school student already uses the concept "subconsciously" (see my comment of 15:24, 21 January 2009, below); he just doesn't know it's called this way. -- Army1987 – Deeds, not words. 14:49, 24 January 2009 (UTC)

(<--)I'm aware of these guidelines and share their intentions. But, perhaps, let's not dig our heels in guidelines and their exegesis. Rather, where precisely does the lead tease the reader or does not explain important facts? Does it contain any lies?

The lead section image is a (pretty brief and in several senses sketchy) explanation of the two operations. Explaining them in more detail and with more care later is perfectly appropriate (and without alternative). Jakob.scholbach (talk) 15:10, 24 January 2009 (UTC)

• Oppose For the same reason as Shoemaker's Holiday. I think the lead should be readable by a reader coming just out or still being in a high school. The first paragraph of the lead for example:

<quote>A vector space is a mathematical structure formed by a collection of objects, called vectors, that may be scaled and added. In a vector space these two operations adhere to a number of axioms that generalize common properties of pairs and triples of real numbers, and of vectors in the plane or three-dimensional Euclidean space. </quote>

• • "scaled" should be "multiplied by a number".
  • "generalize common properties of pairs and triples of real numbers" should be "generalize common properties of numbers"
  • "vectors in the plane or three-dimensional Euclidean space" should be "vectors in geometry"

Of course footnote and link to the exact mathematical meaning can be added but the non mathematician should not be repelled by the lead.

• • The first picture is awful. The caption carries much too much information which is not needed at this point. The vectors are missing arrows. I prefer fig 2 but maybe without coordinate axis.
  • The first example presented in the example section should be a very simple and wellknown example such as vectors in physics (addition of velocities or forces) or in 2D geometry. That the first example mixes functions and fields is a very bad idea. (Fields are simple for those who know what it is but exhausting for the layman which has to click on the link to understand what it could be). Vb 08:30, 20 January 2009 (UTC) —Preceding unsigned comment added by 213.168.116.224 (talk)
    • Strongly disagree with the above comments Let me note that you must know what a vector is before you learn about vector spaces. If you don't know what a vector is, I don't see the point in learning about vector spaces, anyhow! I agree that university students should understand it but certainly not high school students who are not in their final year. In my experience, I have seen people who simply don't understand the "abstract concept" of a vector. Indeed, it is something different compared to what people would think, i.e counterintuitive, because heaps of people think mathematics is centered about number theory. On the other hand, vectors are a very natural concept. So for someone to understand them, they shouldn't sit at the computer and read the screen blankly: certainly not. They should take the time for thinking about what is written in the article and connect it with their prior knowledge. Learning to think is also one of those concepts that must be conquered at some point; being introduced to vector spaces may teach this important skill.

• • • On the same note, "scaled" should not be "multiplied by a number". Using the latter may lead to confusions because many people think that the only numbers are integers; some know of the existence of complex numbers: but there are other fields. We must not "put down" the generality of vector spaces. Similarly, I disagree with your second point. Many people (including those who know calculus) think that R is not a vector space. They simply think "space" must be something complex: i.e something having high dimension. So "pairs and triples of real numbers" may allow them to relate to their prior knowledge of vectors. Again, I don't really like the idea of representing vectors as arrows. Although in a sense, the represent direction, we could write them in bold face. Note that using the convention of "arrows" will lead to difficulties when we consider an arbitrary field over itself. Because in this case, an element of the vector space is a vector, iff it is a scalar. Your last point is already taken care of. Examples of fields are given, and it is also noted that the scalars belong to the field. I think that it all one needs to know.

• • • Cheers, PST 09:40, 20 January 2009 (UTC)

• • • The 2D geometry example is in the 'motivation' section. Do you think it's out of place there? Algebraist 09:32, 20 January 2009 (UTC)

Response to 213.168.116.224: thanks for your points. I'll try to adress them as much as possible, without removing or falsifying facts. For example, "generalize common properties of numbers" does not give the right intuition (also not for a high-school person). I'll respond in more detail soon. Jakob.scholbach (talk) 10:42, 20 January 2009 (UTC)

The whole first section of the lead is reworked. Do you think this is OK now? I will brush over the motivation section next, focussing on concrete physical examples (forces or velocities). Also the lead section image is now clearer. (I uploaded File:Vector_addition_and_scaling.svg, but somehow the labels of the arrows have black background. Does anybody know how to fix this? I used inkscape.) Jakob.scholbach (talk) 18:48, 20 January 2009 (UTC)

I fixed the problem. I had run into this before and was supplied with the fix: there's a problem with text sometimes, so you need to convert all the text to a path (in Inkscape, go to "Path -> Object to path"). I also fixed the scalar multiplication symbol. RobHar (talk) 14:39, 21 January 2009 (UTC)

The motivation section is now rewritten and should be accessible. Please check it out. Jakob.scholbach (talk) 07:56, 23 January 2009 (UTC)

• Comment I would like to respond here to Point-set topologist. I am a math teacher in high school and the topic "vector space" has a short section in the math book we use. The topic is therefore not only usefull to univesity students but also at high school. At this level, even if the students 'should' know what a real number is, it is often the case that they dislike to use such term they consider pedantic. The same is true for Euclidian geometry. Today students don't even know Euclide at his axioms but all still do Euclidean geometry without knowing it: they simply call it geometry. Thus I believe it is superfluous to tell the reader this is here Euclidean geometry. However a foot note or a link such as geometry would be enough. I think the arrows should not be used for labelling vectors but in the first picture arrowheads should be used! (just as in Fig. 2) Vb (talk) 13:42, 20 January 2009 (UTC)
  • A link such as geometry would violate WP:EGG. -- Army1987 – Deeds, not words. 17:33, 20 January 2009 (UTC)

To Point-set topologist: But this is the article about vectors in general. Vector (mathematics) redirects here, as it would make little sense to have a separate article for that. So, as for "you must know what a vector is before you learn about vector spaces", it's this article that should explain "what a vector is", at least until someone creates a separate article Vector (mathematics) (which would be quite pointless IMO). As for "scaled" vs "multiplied by a number", the current version of the lead has the same problem, too. But I'm trying to address it. -- Army1987 – Deeds, not words. 13:55, 20 January 2009 (UTC)

I want to note that there are intelligent high school students who would appreciate the notion of a vector space. But most do not. You see, some high school students do maths for the sake of doing mathematics and certainly would think of vectors as doing "calculations". Indeed, in most high schools, vectors are merely added and multiplied (dot/inner product) without actually understanding them. Of course, they are also taught in the context of physics. But vectors are not physics. They are certainly used in physics but we should not give the wrong impression. As you are a high school teacher, you will probably know high school students better than me (I haven't seen the animals since I was at school :)) but I am a bit dubious that a high school textbook has a short section on vector spaces. Could you note the name of the book? Does it refer to fields (the algebraic structure)? Not that I am saying that students can't handle this, but I would be interested to know the book.

Without knowing about vectors in 2-space or 3-space, the concept of a vector space is not very intuitive. I.e, you won't get a feel for vector spaces. That is what I meant.

Cheers, PST 18:28, 20 January 2009 (UTC)

The book we use here in Germany is the very standard Lambacher-Schweizer Analytische Geometrie und lineare Algebra. Examples of vector spaces are presented which are not (as in the rest of the book) Euclidean vectors (although the word Euclidean vector never appears in the book). The example I discuss with my student is the space of polynoms with a given grade. Other space are cited but they are in fact simply R^n with n>3. I think the student who wants to learn bout this topic must not be a paricularly intelligent one but simply a pupil who wants to prepare a talk in order to improve his participation note! This student does not really remember what is "commutativity" or what is the difference between a real and a rational number even if he uses this concept in the day-to-day practice. About the recent modification of the lead. The first Figure and its caption is now good and explain quite well what is meant by adding and scaling. In my opinion the wording "multiplied by a number" should replace "scaled" and not be added to it. However I think that, now that the picture has been improved, the word "scaled" may be good enough. On the other hand, I agree with the comments which appeared on the talk page and think that putting the physics examples in the first paragraph of the lead enhance much too much the importance of this POV on he meaning of vectors. What I suggest is to put this example (which should be much expanded) as the first example within the example section. Moreover I think the interpretation of vectors as translations is missing in the example section. Please also refrain to use the word Field in the first example. Of course a field is a structure which is simpler than vector space but not for pupils! Vb (talk) 10:28, 21 January 2009 (UTC)

• Comment I have some doubts about the history -- it doesn't match the way I learned it. As I understood it, the basic motivation for the concept was a desire to be able to do arithmetic in Euclidean N-space. The first attempts tried to take off from the fact that complex numbers allow arithmetic in 2-space. However, attempts to generalize complex arithmetic to higher dimensions broke down on the fact that there is no natural multiplication operation. Hamilton's quaternions were a way to make it work for N=4, but gave up on some critical properties and still didn't generalize. Finally in the late 1800s mathematicians realized that the proper thing to do is to give up on a VxV->V type of multiplication, and settle for two other types of multiplication, scalar mult and the inner product. Looie496 (talk) 02:42, 21 January 2009 (UTC)

Although the modern notion of a vector in physics owes much to Hamilton, Gibbs, and Heaviside, the notion of a vector space does not appear to share exactly the same lineage. Hermann Grassmann first introduced the general notion in the same year as Hamilton introduced his quaternions, but it was several decades until Peano and Whitehead ultimately formalized Grassmann's notion of a vector space. My understanding of the genesis of the modern idea of a vector space comes chiefly from the historical notes to Bourbaki's Algebra. To quote: "Peano, one of the creators of the axiomatic method and also one of the first mathematicians fully to appreciate the work of Grassmann, gives as early as 1888 (...) the axiomatic definition of vector spaces (...) over the field of real numbers ..." siℓℓy rabbit (talk) 03:03, 21 January 2009 (UTC)

I agree with Silly rabbit; which also concurs with the historical sources cited in the section. Looie, can you present a reference that backs up your point? Jakob.scholbach (talk) 17:12, 23 January 2009 (UTC)

I'm speaking from memory—but quaternion#History gives an account that more or less matches what I wrote, I think. Looie496 (talk) 20:20, 23 January 2009 (UTC)

What you wrote (and what is in the quaternion article) is correct as far as it goes, but as Silly rabbit points out, that's the history of vectors as in physics and vector calculus. The history of vectors as in algebraic vector spaces is somewhat different. Algebraist 20:49, 23 January 2009 (UTC)

To PST re "Without knowing about vectors in 2-space or 3-space, the concept of a vector space is not very intuitive.": I am playing a strategy game in which you have four kinds of resources: wood, clay, iron and wheat. It is quite intuitive that you can add "100 units of wood, 50 clay, 23 iron and 75 wheat" and "15 wood, 40 clay, 70 iron and 30 wheat" to get "175 wood, 90 clay, 93 iron and 105 wheat", or multiply the latter by two to get "30 wood, 80 clay, 140 iron, and 60 wheat". This is not really a vector space because you can only have an integer amount of any resource, but it wouldn't be any less intuitive if you could. -- Army1987 – Deeds, not words. 15:24, 21 January 2009 (UTC)

• Comments on the lead. After being asked to comment on the lead, I read the article quite closely this afternoon. WP:LEAD states: "The lead serves both as an introduction to the article below and as a short, independent summary of the important aspects of the article's topic." In articles on advanced mathematics (and even though vector spaces are extremely standard mathematics, the abstraction involved is advanced), achieving both of these goals in four paragraphs, while remaining as accessible as possible is a very difficult task. However, Wikipedia is not a textbook: it is not the purpose of the lead to teach readers what a vector space is, but to whet their appetite to learn more.

At the moment the lead does not adequately summarize the article: it focuses unnecessarily on forces, and on applications in analysis, without covering adequately the fundamental role of vector spaces in linear algebra. I realise that my view may contradict the view of some other editors, who are concentrating on making the lead easier to understand for the lay reader. That is a painful aspect of FAC: it is impossible to please everyone.

However, as a positive suggestion, should this FAC fail, one solution to the difficulty of both summarizing the article and providing an accessible introduction is to spin out the lead (summary style) to an "Introduction to vector spaces" article (or possibly an "Introduction to linear algebra"). This would make it easier to make the lead an encyclopedic summary of the topic , while providing an entrypoint for motivated high school students and similar readers. Geometry guy 21:56, 24 January 2009 (UTC)

• Comments on the prose. In the spirit that FAC is a painful experience, the main thing that struck me on my read-through is that the prose is woeful. Sorry, I should say there is lots of good stuff, but there are patches that are painful to read, even for someone who is extremely familiar with vector spaces and knows what the prose is trying to say. I found myself banging my head in a Quasimodo-like experience of "The prose! The prose!". This may be as painful for article editors to read as it is for me to say. So, let me add that tremendous work has gone into this article and it is well on its way to being featured, following the path of Group (mathematics), which was featured thanks to the drive of the same main editor. Bravo! I wish I had the time and energy to do as much to help. I would be happy to copyedit the article, but realistically, I can't do that for at least two weeks, so let me highlight some issues. In any case I'd rather copyedit the article in a minor way rather than make big changes.
  • Encyclopedic language is not flowery. I can see that considerable effort has been made to use encyclopedic rather than textbook prose. However, encyclopedic language is neither flowery nor convoluted. Don't say "keystone" when you mean "central", "in the guise of" when you mean "as", "employed" when you mean "used". "Achieved" is shorter than "accomplished" and "provide" is more widely understood than "furnish". Other flowery usages include "envisaged", "encompasses", "conception", ...
  • Use bland adverbs sparingly. "Historically", "today", "actually", "notably", "usually", "particularly", "especially", "essentially", "completely", "roughly", "simply". These can often be omitted, or replaced by prose which centres the point.
  • Don't split infinitives needlessly. Sometimes they need to be split, but in most cases they don't. For instance, I would replace "To simultaneously encompass", by "To cover" or "To include".
  • Avoid editorial opinion. Whenever you use an adjective of opinion, or comment on the importance of something, it is helpful to ask the question "according to whom?". Then you can decide whether to provide a source, or to rephrase. Adjectives used here include "important", "crucial", "frequent", "fundamental", "suitable", "useful", ...
  • Avoid long noun phrases. They tend to lead to bad prose: "Resolving a periodic function into a sum of trigonometric functions forms a Fourier series, a technique much used in physics and engineering." is an example.
  • Omit needless words. "certain" is usually not needed and "call for the ability to decide" is wordy.
  • Use a consistent English variant: I see both "analog" and "honor" (American) and "idealised" (British).

Finally, resist the temptation to tell the reader how to look at the subject. This happens at the beginning of quite a few sections:

• "Vector spaces have manifold applications as they occur in many circumstances, namely wherever functions with values in some field are involved."
• "'Measuring' vectors is a frequent need"
• "Bases reveal the structure of vector spaces in a concise way"
• "The counterpart to subspaces are quotient vector spaces." (Only makes sense to those who already know what it means.)

Geometry guy 23:15, 24 January 2009 (UTC)

• Further comments.
  • Why does the history stop at c. 1920? The relations to set theory could be discussed here, as could modern developments in homological algebra, Hilbert spaces (quantum mechanics), Banach spaces (Gowers).
  • The category of vector spaces is not boring, nor well understood, even if the isomorphism classes of objects are. See Quiver (mathematics).
  • The determinant of a linear map is not defined, but used.
  • The motivation section is poorly written: if "force" is a motivation, why not explain that addition of vectors corresponds to combining forces?
  • Hamel and Schauder bases are not clearly delineated. Too much effort is expended making the definition of Hamel basis apply in the infinite dimensional case. Further, refering to the existence of Hamel bases (and hence the axiom of choice) as "fact" is point of view.

Despite all my complaints, however, I must say that Wikipedia readers will be extremely fortunate to find such a comprehensive article on such an important concept. Geometry guy 23:37, 24 January 2009 (UTC)

• Withdrawal of the nomination: I would take the opportunity to thank again everybody involved in the recent development of the article, be it in form of critial comments at FAC, at the talk page or working on the article itself. I think it is better to amend all these things carefully (even if I believe the article has improved substantially, and beyond the level I was even thinking of prior to FAC) and especially without haste. To do so, I'd like to withdraw the nomination. I'll work on the issues raised above and may renominate it for FAC at some point. Jakob.scholbach (talk) 08:07, 25 January 2009 (UTC)

Some feedback

I'm putting my response here for the ease of edit (for me and others). By categorical point of view, I was thinking of, for example, the fact that the category of finite-dimensional vector spaces is equivalent to the category of matrices (see Equivalence_(category_theory)). This is very important because it explains, for example, why in linear algebra one essentially doesn't have to study vector spaces as much as matrices. (I also think the cat of finite-dimensional vector spaces is equivalent to that of finite sets.) Also, one may start with a quotient map (i.e., a surjective linear map) instead of quotient spaces and use the universality to show this definition is essentially equivalent to the more usual one. The view points such as the above are abstract but are indispensable if one wants to study vector spaces seriously. On the other hand, I don't think, as the article currently does, mentioning the category of vector spaces is additive is important, for it is very trivial. It is important to mention the applications of isomorphisms theorem rather than how to prove them.

Next, about annihilators. (This is an important concept and the article has to discuss it) I think I was getting at is that the possibility of defining a bilinear form (or sesquilinear one) on a vector space. When studying vector spaces or related stuff in application, bilinear forms defined on them are often useful and indispensable. An inner product is one example, of course, but it doesn't scale well to infinite-dimensional vector spaces (which may not have topology, like infinite-dimensional Lie algebras). So, one also uses natural pairs for V x V^*. (Though this isn't quite a bilinear form.) Anyway, my point is that we need a discussion on bilinear forms (probably a whole section on it). A basis can be chosen according to such a form, and actually that's often what one does; e.g., orthonormal basis. (I just noticed the article doesn't even mention dual basis, which is an important concept.)

Finally, on the balance. Yes, the article is fairly lengthy already, but I think we can make a significant cut by eliminating stuff on trivial facts or some linear algebra materials such as determinants. Doing that would likely diminish accessibility (and thus usefulness) of the article for a first-time learner of vector spaces. But that's something we can afford since the focus of the article should be on important topics not trivial ones. —Preceding unsigned comment added by TakuyaMurata (talk • contribs) 14:09, 25 January 2009 (UTC)

The category of finite dimensional vector spaces is not equivalent to the category of finite sets: the former has many more morphisms, and the latter is not even an additive category. Geometry guy 14:46, 25 January 2009 (UTC)

I think the original poster may have in mind the fact that all vector spaces are free objects. This is not correct either, however, because the choice of a basis isn't natural. Ozob (talk) 21:54, 25 January 2009 (UTC)

The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.