Talk:Vector space/Archive 4

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Comments from the recently withdrawn (or failed) FAC

I'm copying some comments from the recently withdrawn FAC, in order to discuss them here (and/or address them). Jakob.scholbach (talk) 21:16, 26 January 2009 (UTC)

Comments from Geometry guy

The following comment was copied from Wikipedia:Featured_article_candidates/Vector_space. Jakob.scholbach (talk) 21:16, 26 January 2009 (UTC)

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• 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)

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• Ad 2: Actually, I don't quite concur with "resist the temptation to tell the reader how to look at the subject". Obviously, we have to tell it rightly. But assuming we are able to do it rightly, I can't see why we should not do it. I mean, certainly wording here and there can be changed, but what is wrong, e.g., with "Bases reveal the structure of vector spaces in a concise way"? (The 4th point is not great, though, I agree). Giving a prose-style description of some mathematical fact is often conveying more of an intuition than mere definitions and sober statements of facts. This intuition is then fleshed out by the latter.
• Ad 3: History:

• What exactly do you mean by relation to set theory? Things like existence of bases vs. Zorn's lemma? If it's only that, I think that is less of a historical fact (in the sense of a longer development), but just a single mathematical fact (or a few).
• Developments in homological algebra: in what sense were vector spaces (historically) crucial to h.a.?
• Hilbert and Banach spaces: for brevity's sake and also for general considerations, I think historical development of H. and B. spaces should be treated in History of functional analysis, or the corresponding sections in Hilbert space and Banach space.

• As far as I see, the article does not call the category of vector spaces boring or easy. Anyhow, the "degree of boredom" of it should be discussed in the subpage.
• "Too much effort is expended making the definition of Hamel basis apply in the infinite dimensional case": Hm. Do you talk about taking the index set I and denoting the basis vectors by v_ij? I agree that the double index is a bit cumbersome, but I would really not say: "for simplicity, we treat only the case of finite bases". I don't see how to give the (necessary) information that there are infinite-dimensional spaces without introducing infinite (Hamel) bases.
• "fact" is point of view: Hm. It could easily be reworded to "Every v.sp has a basis. This follows from ...", but in what sense would this not be POV? The only really clean way would be "Assuming the axiom of choice or, equivalenlty, L. of Z., every vsp has a basis", but this seems a bit exaggerated, right? Jakob.scholbach (talk) 16:08, 27 January 2009 (UTC)

A review is only a review, and reviewers differ, so there's no reason for you to concur. I will respond with clarifications where I can. Concerning bases, if we want to tell the reader how to think of them, we could equally say "Bases obscure the structure of vector spaces in an ugly way". Indeed, the equivalence of categories with matrices has led one reader to extend that equivalence, erroneously, to the category of finite sets. I agree it is nice to flesh out the bare facts with some intuition, but the latter needs to be backed up by a citation.

On history, set theory, and functional analysis, I can see your point about the latter, but the existence of bases, and the Hahn-Banach theorem play an important role in our intuition about whether the axiom of choice is "true" (and this is disputed). This is an issue which is rather difficult to handle: I dimly recall struggling with it myself at dual space — without choice one might not even be able to prove that the dual space is nontrivial (the article wrongly says "nonempty", but that was probably my gaffe, unfixed for nearly 2 years). Yet we cannot inflict all of this on the hapless reader. The history section may be a better place to discuss it than elsewhere.

You are probably right about homological algebra: the integer coefficient case was probably more dominant historically than the case of a field. These days, however, algebras and modules over algebras are very dominant in homological algebra, as are quivers in representation theory. Stopping the history so early gives a false impression of a sterile subject.

The "boredom" remark was a response to a comment at the FAC, not the article. Gabriel's theorem (1972) shows that almost all linear algebra problems are very hard, including the classification of two endomorphisms, or five subspaces.

As far as I am aware, Hamel bases are essentially useless in the infinite dimensional setting (this is related to the choice needed to show they exist). I'm not sure either how the article should respond to this, but obscuring the meaning of a Hamel basis for the sake of generality may not be the best way. Geometry guy 21:53, 27 January 2009 (UTC)

Perhaps this is fishy, but in a polynomial ring the basis given by the monomials is quite handy. I faintly remember Euler was a sceptic of Fourier transforms--he wondered how a uncountably-diml. space could be "generated" by a countable basis. So, a clear-cut notion of bases also in the infinite diml case seems noteworthy. Jakob.scholbach (talk) 21:58, 28 January 2009 (UTC)

You are right, polynomial algebras are a good motivation for "the" vector space with a countably infinite basis. Do you know a good example in the uncountable case? Geometry guy 22:06, 28 January 2009 (UTC)

(<--)How about the group algebra C[C] or the like? (It's not terribly all over the place, though). Jakob.scholbach (talk) 22:10, 28 January 2009 (UTC)

That's somewhat artificial unless you take into account the topology on C. But then it's probably countably generated as a topological vector space (by all monomials x_α, α∈Q + iQ). Ozob (talk) 01:40, 29 January 2009 (UTC)

Right. Jakob.scholbach (talk) 09:33, 29 January 2009 (UTC)

Perhaps it's not too wrong to say: any object of somehow uncountable nature is obtained via limiting processes (i.e., topology)? Jakob.scholbach (talk) 09:35, 29 January 2009 (UTC)

If you are after a real world example of a nonseparable Hilbert spaces where the use of a basis is useful you just have to look at Quantum field theory. The basis states of QFT are of the form $\left|k_{1},...,k_{n}\right\rangle$, where the k_i are momenta. The corresponding Hilbert space is (the closure) of the span of these states. Since all of these state are orthoganal and the momenta live in a continuous space (as long as the QFT lives on a non-compact manifold) the Hilbert space is clearly nonseperable. Yet, the only sensible way of talking about it is in terms of its basis. (TimothyRias (talk) 10:40, 29 January 2009 (UTC))

I guess you are talking about a base in the topological sense? We try to find an example of a useful, not totally artificial (as opposed to $\oplus _{i\in I}\mathbb {R}$ for some set I) vector space of uncountable (Hamel) dimension, a base of which can be explicitly given. Jakob.scholbach (talk) 16:19, 29 January 2009 (UTC)

Even deleting "the closure" parenthesis, the physical meaning is unclear, given our lack of understanding of physics at the Planck scale. Geometry guy 19:55, 29 January 2009 (UTC)

This is philosophy, but I think Jakob is right. We can only comprehend uncountable sets by viewing many of their elements as being very close together. That intuition inevitably has a topological flavour. Geometry guy 19:55, 29 January 2009 (UTC)

If I follow the points here correctly then such an example is considered in Loop Quantum Gravity. There the Hilbert space of functionals over (generalized) connections on a 3-manifold is defined using an inverse limit. This is non-seperable. It decomposes into subspaces labeled by all embedded networks. —Preceding unsigned comment added by 212.76.37.234 (talk) 23:06, 16 May 2010 (UTC)

Comments by TakuyaMurata

The following comment was copied from Wikipedia:Featured_article_candidates/Vector_space. Jakob.scholbach (talk) 21:16, 26 January 2009 (UTC)

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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)

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I feel the need to repeat (although I'm sure it is well understood) that the category of finite dimensional vector spaces is not equivalent to the category of finite sets. Geometry guy 21:43, 26 January 2009 (UTC)

Certainly not if the morphisms between finite sets are the usual ones: i.e., just functions from the first set to the second. However, one can make sense of Taku's comment: given a field k and a finite set S, one can consider the free k-module k[S] on S, and then if we define Hom(S,T) to be Hom_k(k[S],k[T]), then that category is indeed equivalent to the usual category of k-fdvs. I have heard category theorists describe things this way before, so it is probably what Taku had in mind. Plclark (talk) 07:15, 28 January 2009 (UTC)

It is unlikely that he meant that. If he did, he should have been more explicit and actually described each category given. But I agree with what you say. --PST 17:20, 28 January 2009 (UTC)

• I'm puzzled by the following: "An inner product is one example, of course, but it doesn't scale well to infinite-dimensional vector spaces". Surely Taku has heard of Hilbert spaces or, what is more directly relevant, "pre-Hilbert spaces"; the latter is precisely an infinite dimensional space with an inner product. Of course, the inner product then defines the topology, which may or may not be complete. siℓℓy rabbit (talk) 22:33, 26 January 2009 (UTC)
• Taku, could you please not cut stuff from the article? As a good article, vector space does stay focussed and has an appropriate amount of content. In fact, it is missing content and accessibility (it seems). So rather, you should try adding some stuff to the article rather than deleting. And by the way, determinants are relevant to vector spaces. The determinant of a linear map between vector spaces is 0 iff the map is not an isomorphism. And it is fairly clear that the category of finite sets is not equivalent to the category of finite dimensional vector spaces. Surely, you know what a category and a functor are (if you know both definitions, I can't see how this is non-trivial)? --PST 08:58, 27 January 2009 (UTC)
  Taku: another comment. I don't want to sound rude, but it seems that you are not familiar with the FA guidelines. We must make technical articles accessible (please see the link) and we can't randomly delete content. Jakob is very familiar with the FA policies and I think that it is a safe bet that he knows what he is doing (when adding the section on determinants). Wikipedia is also not solely for reference work. For example, although I see many articles that I have never heard about every day, I also see some important facts in the lower level articles which I never knew about. It is a question about what we want to do with Wikipedia. In my view, Wikipedia should be aimed at everyone. For instance, suppose you did not know something in computer science and you wanted to learn about it? What if you went to the article and got piled with tecnical info which you didn't understand? Would that do you any good? We all want to learn and by using tecnical terms in subjects we don't know about, we wipe out this wonderful oppurtunity. I hope you feel the same way (for another example, I'm also interested in things apart from mathematics: it does not help me if the articles written use terms beyond my knowledge. It is also an important skill to be able to write technical articles in an accessible manner. Whenever I see such articles, I am always impressed.). --PST 09:49, 27 January 2009 (UTC)

(<-)I also think we must not remove "trivial" (for you!, but this is one the 500 most visited articles, I'm sure it's not only researchers who want to cheer up their minds with a cozy basic WP article) material. On the contrary, we must give an accessible account (which is challenging). If possible, (and this is the 2nd challenge) we can interweave more advanced material in a way such that the reader hardly ever thinks "eh, this is hard stuff".

Equivalence of categories of vsp and matrices: I have to say I disagree a bit with the point of view, linear algebra be all about matrices. The article does state this point, but does not call it an equivalence of categories. The most space I'd give to this point (in this article) is a footnote.

Universal properties of quotients etc. This is something that is not very specific to vsp. Also it will repel 95% of the readers. I would not write about that.

Bilinear forms: there is a brief mention at the tensor product section, but perhaps we can highlight their importance a bit more. I'm not convinced, though, that we have to mention annihilators here. That should go to bilinear form or, at most, to the see also section. Jakob.scholbach (talk) 16:18, 27 January 2009 (UTC)

Comments by Point-set topologist

General comments

• The current intro to the article is weighed too much towards physics. While most people learn about physics before vector spaces, the current discussion is centered towards properties of Euclidean vectors; not vector spaces. I think it would be important to note that vectors have a magnitude and a direction (so in some sense, they "induce" a co-ordinate system). Suprisingly, this does not seem to be mentioned.

• I think it would be good to have a section titled "Vectors in physics" rather than include it in the motivation section. In fact, I think it is unfair to mathematicians to say that physics was the reason for the invention of vector spaces. Vector spaces have so many important applications in mathematics.

• What about noting something on the linear functional and stating that the integral operator is a linear functional from the vector space of continuous real-valued functions on a compact interval, to the vector space R?

More comments later... --PST 12:10, 27 January 2009 (UTC)

I have rewritten the motivation section. As G-guy points out, it's best to work on the lead when everything else is finished. "Vectors in physics": hm, I don't quite know what to write there. I think the current motivation section, just briefly mentioning force and velocity as examples for Euclidean vectors might be sufficient. Integral as a linear functional is already covered (in the distributions section). Jakob.scholbach (talk) 21:44, 28 January 2009 (UTC)

Point-set topologist, I am the author of the edit you recently reverted, which you explained by writing "...although it was done in good faith, the content of the edit did not conform to the article's layout." I do not feel that a bald revert was appropriate, because what I changed was simply not up to snuff. I ask that you restore what I wrote, after which you are welcome to edit my edit so that it reads as you would prefer.

This section of the entry should introduce the reader to how vector spaces relate to other basic algebraic structures. Specifically, a vector space marries an additive abelian group to a field, adding the bridging axioms that characterise modules. It should also mention that a vector space is a variety in the sense of universal algebra, which is perhaps surprising because a field is not. The entry should be consistent with:

http://math.chapman.edu/structuresold/files/Vector_spaces.pdf

Vector spaces are not esoterica; they are arguably the most ubiquitous of algebraic structure having more than one underlying set, and this entry should be written accordingly. I discovered vector spaces via mathematical statistics and economic theory, but freely grant their power in engineering and physics.132.181.160.42 (talk) 00:03, 14 May 2009 (UTC)

To begin, please post at the bottom of the page as this is the convention of Wikipedia. Please see [1]. The link does not serve to argue that my revert was correct; rather it explains that a similar edit has been made in the past. Modules are explained later in the article - the beginning was carefully constructed through hard work, to ensure that it is accessible. To conform with WP:MTAA, it is only appropriate to introduce modules once simple concepts about vector spaces are introduced such as subspace, basis etc... The concept of a module is abstract to most beginners, and are an analogue of "vector space over a field" with "field" replaced by "ring". Up to the point where you have edited, the concept of a field has not been formalized, and therefore it is not appropriate to start talking about rings (otherwise, fields should have been formally defined earlier). I fully understand that you edited with good intentions, but Wikipedia is a complex place. Articles such as this one are GA's and usually are of fine quality. Furthermore, your claim that this article should be consistent with the treatment of vector spaces at some university, is not justified. It is important to understand that Wikipedia is an encyclopedia, and that it should note resemble lecture notes, textbook content etc... I have contacted User:Jakob.scholbach and am leaving it to him and other Wikipedians to decide whether to keep your edit or not. These are the experienced Wikipedians whose decision, I do not doubt, will be justified clearly. On the other hand, as a Wikipedian with little experience, my view is that your edit, although done with good faith, makes the article somewhat redundant. This is not to say that my view is correct. --PST 08:02, 14 May 2009 (UTC)

I fully agree with PST. There are many ways to vector spaces, but the main aim of WP is to make it accessible to people who don't know yet. Talking of modules is helpful only to readers who only know modules (and most of them will, I bet, then also know vsp.), therefore mentioning the connection is done down in the appropriate section. Jakob.scholbach (talk) 08:20, 14 May 2009 (UTC)

Comments from Awickert

The following text is copied from my talk page Jakob.scholbach (talk) 21:39, 16 February 2009 (UTC)

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I saw that you were looking for someone to read it. I don't qualify for the lack of knowledge, but I don't think in mathspeak, so maybe I'll have a crack at the lead and leave you a message here. Awickert (talk) 05:14, 9 February 2009 (UTC)

OK - just from giving it a skim, it seems to be on the dense and rigorous side to me. When I think of a vector space, I think of a set of vectors that define a n-dimensional space that consists of everything those vectors can reach. It's a much more tangible thing to me than a mathematical construct. So maybe something like this (which is rough, and I might be making mathematical terminology atrocities):

"A vector space is the set of all points in space that is accessible by combining multiples of the vectors that describe it."

And then maybe: "Its dimension is defined by the number of vectors in independent directions that constitute it; for example, three mutually perpendicular vectors define a three-dimensional rectangular coordinate system that is often used to describe the three observed spatial dimensions."

I don't know if the suggested sentences are any good, but I would suggest to ground it to the real world right away in some way and tell a reader what it can mean outside of just the formal mathematical definition.

Awickert (talk) 05:23, 9 February 2009 (UTC)

One other dubiously useful comment that could show you how a non-mathematician may use vector spaces as a construct for thinking: when solving problems with large numbers of unknowns and not enough constraints, I always think of the problem in terms of narrowing down an n-dimensional vector space and trying to get it down to a single point (i.e., use basic physics to constrain this, use this empirical relationship for that, can't constrain this axis but can set boundary conditions, etc. etc.). Awickert (talk) 05:27, 9 February 2009 (UTC)

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Could we first focus on the motivation and definition section?I guess the lead will have some overhaul anyway, and it seems best to touch it only at the end... I'd appreciate if you would comment on the definition section. Thanks, Jakob.scholbach (talk) 21:41, 16 February 2009 (UTC)

OK - I'll get to it when I have a chance. I'm going to be pretty busy for the next week. Awickert (talk) 21:45, 16 February 2009 (UTC)

Differential geometry

The second half of the application subsection on differential geometry may need a bit of a rethink. Space-time is not (modelled by) a Riemannian manifold, the Einstein curvature tensor is only part of its curvature, differential forms are (usually) sections of the exterior algebra bundle of the cotangent bundle, not elements of a fibre, and they do not so much "generalize" the dx in calculus, as provide a way to interpret it (for instance dy/dx is a ratio of 1-forms on a 1-manifold).

There is also a question of focus. Linear algebra is used extensively in differential geometry (indeed, modulo the inverse function theorem, which is rapidly brushed under the carpet of intuition, the toolkit of differential geometry consists of little more than linear algebra and the product rule), so what should we select? We need to stay on topic here, and the topic is applications of vector spaces. The Lie algebra tangent to a Lie group seems like a good example to me, but curvature and integration may be a bit of a stretch. Any thoughts? Geometry guy 18:30, 1 March 2009 (UTC)

I may have been misguided by false beliefs/priorities in writing the stuff, but my intention was to somehow give an impression how universal vector spaces are. I wanted to avoid arid statements, instead offering a bouquet of guideposts leading in different directions. (The target audience for these bits of text I wanted to talk to is (advanced) undergrads who may have become bored by pure linear algebra.) It is probably right that the road from vsp to space-time is a bit (too) long, also Riemannian geometry may be far. However, if the "guideposts" idea is taken seriously, a reader willing to make the connection will have to read the referred topics anyway, so we may, I think, come up with kind of overview statements. Also, if "the toolkit of differential geometry consists of little more than linear algebra and the product rule" is right (and I think it's sound) this also means that diff geo is fairly tightly linked to lin alg and vsp, right?

But, as always, if you see a better way, go ahead. In particular, diff. forms should depend on the base point, that's right. I'm not sure how we can convey in one line the thinking of dy/dx as ratio, though. Jakob.scholbach (talk) 20:25, 2 March 2009 (UTC)

I have moved differential forms to the bundle section (and corrected the flaw). Jakob.scholbach (talk) 20:38, 2 March 2009 (UTC)

My intention was to raise questions and contribute to solutions rather than raise criticisms and propose fixes. The widespread usefulness of vector spaces is certainly something to emphasise and I think your priorities are spot on when you say to give specific pointers rather than bland general statements.

Diff. geom. is certainly tightly linked to linear algebra, but linearity and linear maps feature more strongly than vector spaces. One difficulty with the differential geometry applications is that the most interesting vector spaces are spaces of sections of vector bundles (vector fields, differential forms). Would it be feasible/reasonable to reverse the order of the Applications-Generalizations sections? Then more could be said in the differential geometry direction, although it would be tough going even for your advanced undergraduate reader. Geometry guy 22:18, 2 March 2009 (UTC)

Hm, I'm not sure we should reverse the order of the two sections. We could, though, put a word about vector spaces of sections of bundles in the generalization section and refer back to any applications in diff geo. This quickly brings us to sheaves, too. What are the most prominent examples of spaces of sections? (We already have vector fields). Jakob.scholbach (talk) 14:38, 3 March 2009 (UTC)

RFC: Length of 99,022 bytes?

Length of 99,022 bytes? Is this an article of an encyclopedia, or a single article encyclopedia of almost anything what is somehow related to the title of the article? prohlep (talk) 17:46, 5 June 2009 (UTC)

Yet another issue: the definition of the vector spaces is simply erroneous. As soon as you try to feed it into an automated reasoning system, you will understand why.

This error, I teach for my students in Hungary since 1978, is a good test, whether you correctly understand, what the evaluation of a first order formula does mean.

I leave the correction of this error here in the Wikipedia, as an instructive homework for those, who think that they have a firm natural scientific base for discovering the nature.

prohlep (talk) 17:46, 5 June 2009 (UTC)

The length of the article is perfectly OK. Many GA and FA of such a wide topic are this long. What error are you talking about? Jakob.scholbach (talk) 18:26, 5 June 2009 (UTC)

Well, it would not be instructive, if I told the error.

Too many natural scientists and mathematicians are ready to jump over gaps, asif it was not there. The result is, that some of them are ready to identify the nature with one of it's possible natural scientific model. This leads to blind believe in scientific models, what caused many suffer already.

I have just double checked, the error is still there. It is hard to detect, since similar error can be found in every second course book on vector spaces. If the others avoid this error, then it is usually due to an accident, that if the TYPE of the vector space is declared in a different way, then this fundamental error simply CAN NOT occur.

I have just gave the key hint: the error is in connection with the type of the universal algebra, we want to axiomatize as a vector space.

prohlep (talk) 20:48, 5 June 2009 (UTC)

Are you referring to the inconsistency between treating a vector space as a one-sorted structure containing only the vectors and calling scalar multiplication a binary operation? I don't think that's so bad. It may be possible to improve this, but I am not sure it can be done without compromising on the accessibility of the article. We should most definitely not talk about the usual signature of vector spaces, with one unary homothety for each scalar, or at least not early in the article. It's not even so wrong if you consider that there are many-sorted logical frameworks in which one sort is kept fixed. (Actually the only example I know has a fixed sort containing the natural numbers.) --Hans Adler (talk) 21:51, 5 June 2009 (UTC)

• There is no problem here, and no need for an RFC. – Quadell ^(talk) 19:05, 16 June 2009 (UTC)

added reference in "basis and dimension"

Sorry, I don't know how to reference something in the article, but someone could easily add it.

A reference that {1,x^2, x^3, ...} is a basis for the vector space of polynomials can be found in Abstract Algebra with Applications: Volume 1: Vector Spaces and Groups by Spindler, Karlheinz; p.55, Example 3.14.d.

Setitup (talk) 02:13, 9 November 2009 (UTC)

Isn't it necessary to add that in F(t), the field F has to have an infinite number of elements? Otherwise, I'm not sure that {1,x^2, x^3, ...} is a basis for the vector space of polynomials. —Preceding unsigned comment added by Arayamuswiki (talk • contribs) 00:44, 26 January 2010 (UTC)

The article uses the polynomial ring, not the function space of polynomials, so the field does not have to be infinite. JumpDiscont (talk) 07:33, 28 February 2010 (UTC)

last axiom unnecessary?

The "identity element of scalar multiplication" given by 1v = v seems to follow from the other axioms: 1v-1v=0 1(1)v-1v = 0 1(1v-v) = 0 v=1v 220.239.204.210 (talk) 22:46, 15 November 2009 (UTC)

Without that axiom, nothing stops 1v = 0. That invalidates your last step. Ozob (talk) 02:49, 17 November 2009 (UTC)

Is the Second Axiom Necessary?

The "Commutativity of addition" axiom v + w = w + v seems as though it can be derived from the other vector addition axioms:

Given from the inverse axiom u + (-u) = 0,

Proposition: -u + u = 0

Proof:

-u + (-(-u)) = 0 <-- equation 1; inverse axiom; -u plus the inverse of -u equals the zero vector

LHS:

-u + u

= (-u + u) + 0 identity axiom

= (-u + u) + [-u + (-(-u))] applying equation 1/additive inverse axiom

= [(-u + u) + (-u)] + (-(-u)) associativity axiom

= [-u + (u + (-u))] + (-(-u)) associativity axiom

= [-u + 0] + (-(-u)) inverse axiom

= -u + (-(-u)) identity axiom

= 0 inverse axiom

RHS:

0

Therefore, LHS = RHS.

Q.E.D.

--LordofPens (talk) 08:05, 25 January 2010 (UTC)

Commutativity of addition does indeed follow from the other axioms. The article used to mention this. --Zundark (talk) 08:55, 25 January 2010 (UTC)

I was mistaken. That proof does not prove commutivity in general; it only proves commutivity for the additive inverse axiom. However, commutivity in general u + v = v + u follows from the proof above plus the proof for 0 + u = u and the use of most of the axioms. --LordofPens (talk) 04:35, 26 January 2010 (UTC)

Convex analysis and universal algebra

Maybe I'm displaying my ignorance of universal algebra here, but I don't see how one recently added paragraph can be right. It reads:

In the language of universal algebra, a vector space is an algebra over the universal vector space $K^{\infty }$ of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in here (of finite sequences summing to 1), a cone is an algebra over the universal orthant, and a convex set is an algebra over the universal simplex. This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates".

In order to specify a vector space, you need to specify an underlying field. But you can't specify a field in the language of universal algebra because you can't express the existence of multiplicative inverses as an operation. So you have no way of specifying your coefficients. Furthermore, if you could specify your coefficients, then I don't see why one needs to say that vector spaces are an algebra over K^∞.

I have the feeling that you're doing something different, but I'm not sure what that is. What do you mean by, "an algebra over"? Do you mean that a vector space is an algebra of the same type and that it has a homomorphism from K^∞? If not, then that's what I'm confused about. Ozob (talk) 13:18, 26 February 2010 (UTC)

I also don't understand it. Also I think this article should devote one or two short phrases at most to universal algebra. I guess at least half of the recently added material should be moved to universal algebra or convex analysis. Jakob.scholbach (talk) 22:38, 26 February 2010 (UTC)

I'm starting to get some idea what's going on. Grätzer's and Cohn's books don't list vector spaces in the table of contents or in the index, but searching in them for "vector space" in Google Books gets one a little somewhere. I think it's like this: One has a binary operation, vector addition, and if K is the field, then, for each x in K, one has a unary operation m_x corresponding to multiplication by x. One imposes the group conditions on addition, imposes m_{x + y}(v) = m_x(v) + m_y(v) for all x, y, and v (i.e., addition of operations is defined via distributivity and addition in the field), imposes associativity and commutativity of addition, imposes the relation m_xy(v) = m_x(m_y(v)) (i.e., multiplication of operations is defined by associativity and the field multiplication), and imposes commutativity on scalar multiplication. (I may be forgetting some relations, and I may have screwed some of these up.) Once this is done, an object with this type will be a vector space over K. It feels terribly messy, but in a way it's not: Often times when discussing vector spaces, we leave the field in the background. In this setup, the field really is in the background, since it's part of the type, not the algebra.

I still don't see where K^∞ enters the picture, though. Ozob (talk) 03:44, 27 February 2010 (UTC)

Metric Spaces Explicit Reference

I was looking for definitions of vector and metric spaces, and found that vector spaces are a class of metric spaces. If this is correct, shouldn't this be mentioned in the first lines of the article with the corresponding reference for the non specialists?

Thanks —Preceding unsigned comment added by Ercolino (talk • contribs) 15:26, 29 May 2010 (UTC)

It's not correct. Maybe what you were reading about were normed vector spaces. --Zundark (talk) 15:55, 29 May 2010 (UTC)

Small bug?

Hello: (I'm a novice Wikipedian.) I believe I've found a small bug in the matrices section:

x = (x1, x2, ..., xn) |-> (sum, sum, ..., sum)

Shouldn't the xn term be xm?

x = (x1, x2, ..., xm)...

as there are m rows in the matrix.

Thanks,

Myrikhan (talk) 15:30, 11 September 2010 (UTC)Myrikhan

The article looks right

Any m-by-n matrix A gives rise to a linear map from Fⁿ to F^m,

so its a map from Fⁿ hence the coordinate of the source should be (x₁, x₂, ..., x_n). Also note in x ↦ Ax the matrix is on the left hand side so the number of column agrees with dimension of x.--Salix (talk): 16:36, 11 September 2010 (UTC)

The topic of this article is "Vector space". Yet, in the first paragraph alone, there are three separate subjects, only one of which is a direct reference to the subject of the article. The article gets back on topic at the beginning of the second paragraph, only to go off-topic again.

Does everyone realize that it is not those "who already know" the subject that come to these articles to become better educated? Who those who contribute to articles such as this one "PLEASE" write with those people in mind, as compared to trying to include all the related but secondary information. Please stop leaving out the obvious. It is information that is not obvious to those of us who read these articles for new information.

These articles are not for you to show us how much you know. They are to provide accurate and understandable information to the uninformed who are seeking to be informed.

Richard (talk) 02:16, 26 December 2010 (UTC)

First vector wikilink (in lead)

I find it questionable that this GA defines vector in its lead by linking to a dab. After all, this article is the page where vector is formally defined... If that link to the dab is supposed to give some more intuitive notion, it surely fails to do so. Tijfo098 (talk) 11:01, 19 March 2011 (UTC)

Mixed Up

In the definition of a vector space it contains the text: "In the list below, let u, v, w be arbitrary vectors in V, and a, b be scalars in F." However in the list of axioms the vectors u and w, and the scalars a and b do not appear at all. s and n appear to represent scalars, however in the "Respect of scalar multiplication over field's multiplication" and "Identity element of scalar multiplication" axioms, s is a vector. — Preceding unsigned comment added by 69.30.62.114 (talk) 14:34, 17 October 2011 (UTC)

Direct explicit contradiction

First sentence in lead:"A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context." First sentence in Definition:"A vector space over a field F is a set V together with two binary operators that satisfy the eight axioms listed below." BUT in section Algebras over fields. The first sentence reads:"General vector spaces do not possess a multiplication operation." If this isn't a direct contradiction, I don't know what is. -Assuming these are not actually contradictory, there is some context that needs to be added, some qualification made for this to be intelligible to us mere mortals. Could it be that the multiplication operation that they do not posess is vector by vector multiplication? I suspect so. Could someone modify this so it makes sense? I'm unqualified. Thanks!71.31.147.72 (talk) 17:43, 23 November 2011 (UTC)

There is no contradiction here. The two binary operations the definition talks about are vector addition and multiplication of a vector by a scalar. This is explained in some detail immediately after the definition. The statement in the algebra section refers to the non-existence of a multiplication of two vectors in a general vector space. Jakob.scholbach (talk) 07:45, 24 November 2011 (UTC)

Excellent article

This article is an excellent article and is a benchmark for all other articles to follow suit. PsiEpsilon (talk) 12:13, 1 January 2012 (UTC)

Picture in lead

I think the picture in the lead showing addition and scaling of vectors is nice but grossly out of place. It should be moved lower, or replaced/removed. Rschwieb (talk) 19:23, 9 April 2012 (UTC)

It would be nice if you could propose an alternative image. My impression is that the current one is the least worst, based on earlier revisions. Sławomir Biały (talk) 12:40, 14 April 2012 (UTC)

It's an excellent diagram for visualizing vector operations, but for someone who is learning what a vector is, springing vector operations at the start is not helpful. A single diagram with two or three labeled vectors captioned "two/three vectors" would be a good replacement. We'd like to depict that vectors are like arrows with varieties of lengths and directions. Two of the vectors could have different sources, two be different lengths, and two be in different directions. Rschwieb (talk) 13:05, 14 April 2012 (UTC)

That seems reasonable overall. (Although perhaps not having different sources?) With a suitable caption, I could see that as being more informative to a casual reader than the current image. This was in fact a big issue at the FAC. Sławomir Biały (talk) 12:19, 18 April 2012 (UTC)

I just didn't want to fool readers into thinking all vectors begin at the origin, so that is what I meant by the source comment. It is not a huge deal though, if the new picture has this minor problem. I have no skill to enact any of the changes to the diagram... any good recommendations? Rschwieb (talk) 13:20, 18 April 2012 (UTC)

Complex numbers example

I believe that this section recently added to the article is misleading. Instead of shedding light on the definition of a vector space, it actually obscures that definition. In general, it makes no sense to multiple two vectors. However, this supposed example not only introduces the notion of "multiplication" of two vectors if those vectors happen to be complex numbers, but it alludes to multiplication being well-defined in a general vector space (in general only the operations of addition and homothety are available in a vector space though). The actual (real) vector space structure of the complex numbers is identical to that in the second example, so this adds nothing helpful to elucidate the definition, and only obscures it by introducing structure that is not part of that definition. Sławomir Biały (talk) 13:53, 22 April 2012 (UTC)

(edit conflict)I fully agree with Slawomir. The IP's repeated argument seems to pertain to that the example of the vector space of complex numbers sheds light on the characteristics of a vector space because it includes a different notion of vector multiplication(!) than what is traditionally defined as a scalar product. However, the IP seems to miss the point that a scalar product is not the same as a scalar multiplication; the former is a multiplication between vectors, while the latter is a multiplication between a scalar (from a field F) and a vector (from the vector space over F). Nageh (talk) 14:01, 22 April 2012 (UTC)

I am the guy with IP 177.41.12.4. I am an occasional contributor. I think editors should be tolerant to what people from other backgrounds think and see when they take the time to contribute. As an occasional contributor, I don't like when people just remove content from others on the basis of their personal vision/opinion. They should at least create a talk so that the anonymous guy who made the change can argue afterwards when he comes back at some point if he comes back. Please remember that the idea of wikipedia, as far as I understand, is to have as many people with good technical background contributing as possible. So the editor should not have the attitude of seeing an anonymous contribution and just because he doesn't know the person conclude that the person knows less than him. When I see that text, I see the guy has technical background. To the point: yes, you can multiply vectors in a more abstract sense than you understood, Slabwerk, and you can also divide them. See articles on quaternions. The idea of Hamilton when he invented them was precisely to include an operation to _divide_ vectors. And the complex numbers are the door to this kind of thinking, which is different from yours. So if I had to remove an example I would remove the second one, which for me, as a physicist, is very abstract and does not convey any intuition. Maybe a mathematician would disagree, so I would keep it as well. You see the kind of thinking that makes wikipedia better? Respect to the contributions of others. An editor cannot be so picky. Adieu. — Preceding unsigned comment added by 177.41.12.4 (talk) 14:07, 22 April 2012 (UTC)

No you can't multiply elements of a vector space. That requires additional structure which is not the subject of this article. Sławomir Biały (talk) 14:15, 22 April 2012 (UTC)

To try to explain the issue for the IPs in another way: the complex numbers are extremely important as a model of rotations, etc. They are also a nice example of a vector space. However, the idea of the complex number multiplication as a model of rotation is a fundamentally different issue than the model of the complex numbers as a vector space; in particular, the sentence "Instead of making the product between two vectors as we know it for a common spatial vector, we create the rule ..." is incredibly misleading -- it distracts from the vector space properties of complex numbers to discuss some other (very interesting) properties of complex numbers that come from their being something other than just a vector space. The complex number product is a fascinating, important object, but not as an introductory example of a vector space. Replacing complex multiplication with the multiplication of a complex by a real number would be more appropriate, for example. --Joel B. Lewis (talk) 14:23, 22 April 2012 (UTC)

Except that complex numbers under real homotheties is essentially identical to the ordered pairs example. Sławomir Biały (talk) 14:33, 22 April 2012 (UTC)

I don't see the problem: it's an extremely common problem for students learning about vector spaces for the first time that they don't realize different objects (R^2; complex numbers; degree <=1 polynomials over R with a single variable) are really the same as vector spaces. It seems to me that having the same example twice in different guises could be very helpful for clarifying this issue. --Joel B. Lewis (talk) 14:44, 22 April 2012 (UTC)

Remember, this is in the definition section, not the examples section. There is ample space elsewhere in the article to mention the example of complex numbers (e.g., under field extensions). The examples preceding the definition should be absolutely the minimal set of examples needed for a reader to understand the definition. Too many examples (especially ones that are the same example) will be counterproductive. Sławomir Biały (talk) 14:50, 22 April 2012 (UTC)

Ok, yes, I agree. It would fit much better in the subsequent "Other Examples" section (which should probably also be made much more gentle.) --Joel B. Lewis (talk) 16:20, 22 April 2012 (UTC)

To the IP user: yes you may define a multiplication between two vectors of a vector space. But, as soon you do that your vector space becomes a richer structure which is more than a vector space. Depending on the properties of this multiplication, this richer structure may be called normed vector space or Euclidean vector space, if the multiplication is the inner product, algebra over a field for the example under discussion. There are more exotic multiplications on vector fields, like that of Lie algebra, widely used in physics. It may be relevant to mention all this in this article, but only after the definition of a vector space, and not as a pure example of vector space. As the example has been introduced, it is not acceptable for an encyclopedia, because it induces, for the non expert reader, a confusion between vector space and algebra over a field. D.Lazard (talk) 15:52, 22 April 2012 (UTC)

I agree with Sławomir Biały regarding the IP's edit, but Joel B. Lewis has a point nevertheless. This article (and even the external main article) seems to feature only "abstract" examples of vector spaces and I don't quite see why can't provide at least more concrete like R^2, R^3 or the complex numbers.--Kmhkmh (talk) 10:48, 23 April 2012 (UTC)

I think the examples already given should be expanded and made a little more concrete. A paragraph could be added to the coordinate spaces section at the beginning mentioning R^3, and one should be added to the beginning of the field extension section mentioning the complex numbers. Sławomir Biały (talk) 11:44, 23 April 2012 (UTC)

From the edit summaries, the editor at 177.41.12.4 seems to think Slawomir is working alone. Let me express here that I agree with the reasons given by Slawomir for not using this example. The example itself is not bad, it's just misplaced here. Rschwieb (talk) 13:47, 23 April 2012 (UTC)

I completely agree with Slawomir. --Txebixev (talk) 11:45, 27 April 2012 (UTC)

Errors in the definition of basis of a vector space

1) The section Bases and dimension begins as follows:

"Bases reveal the structure of vector spaces in a concise way. A basis is defined as a (finite or infinite) set B = {vi}i ∈ I of vectors vi indexed by some index set I that spans the whole space, and is minimal with this property."

But a basis has nothing to do with an index set -- and the index set is not used in this definition, either, beyond the vacuous statement that the basis vectors are indexed by it.

This should be deleted.

2) Soon after, this section reads:

" Minimality, on the other hand, is made formal by requiring B to be linearly independent."

But linear independence has nothing at all to do with making formal the statement that a basis is a set of vectors minimal with respect to the property of spanning the vector space!

3) On the other hand, an equivalent definition of basis that deserves equal standing with the minimality condition is that a basis is a set of vectors that is linearly independent and is maximal with respect to this property.

4) A basis should be defined as a set of vectors satisfying either the minimality condition or the maximality condition, which are logically equivalent.Daqu (talk) 04:57, 27 August 2012 (UTC)

Normally, a basis is defined by using neither minimality or maximality (which are vague and would require further definition) but simply by the two requirements that a)the basis spans the whole vector space. b)The basis elements are linearly independent. In a way, linear independence can indeed be seen as way to formally implement the minimality requirement.TR 07:41, 27 August 2012 (UTC)

I think there is a good point here in that the definition of a basis does not need the index set. It's understandable that it appears, because in linear algebra we are almost always using an ordered basis which does require indexing. Maybe that distinction could be sorted out, if it isn't already somewhere else in the text. Rschwieb (talk) 13:11, 27 August 2012 (UTC)

Can we just define a basis to be "a subset of a vector space that is linearly independent and spans the whole space". Note a basis is the empty set if and only if the space is zero. -- Taku (talk) 13:29, 27 August 2012 (UTC)

We could, but that would involve getting rid of all the helpful motivating text in that paragraph or two. The index set is there not because it's necessary for the definition (it's neither required nor is its inclusion an "error") but because it introduces the notation used in the following equations. This section has a link to the main article basis which is where things need to be in full detail. --JBL (talk) 13:37, 27 August 2012 (UTC)

I don't think it involves that at all. In fact I just did it. Its inclusion was an error because we are talking about two things as if they were one thing, but that is no longer the case. I retained the existing ordered basis for convenience throughout the rest of the section. Rschwieb (talk) 14:00, 27 August 2012 (UTC)

I prefer TR's version -- the issue of basis ordering is at best a minor technical question, which you've now placed so as to make it look like it's of the utmost importance. (Indeed, the are now two long sentences separating the use of the word "span" from the explanation of what it means!) If you insist on making a comment about it (which still seems totally unnecessary to me) then a footnote seems like a good way to go about it. Actually if there were a way to write it without mentioning the words "index set" but still allowing the use of the notation, that would be good to. --JBL (talk) 19:21, 27 August 2012 (UTC)

Being seasoned with the concepts, you would naturally find it minor. However, this is exactly the sort of misassumption teachers have and exactly the sort of snag I see consternating undergraduates from time to time. It is hadly "far away": one sentence (15 odd words) is about "unordered bases", and the other "long sentence" you are referring to says "we'll just use ordered basis notation here". Your last comment about "trying to do it without an index set but still allowing the use of the notation" is fantastic and is exactly the idea I had in mind with those edits. Rschwieb (talk) 20:43, 27 August 2012 (UTC)

Really the indexed/ordered versus unordered thing is a side issue I think. In practice, bases almost always have an index set. This is so that expressions like $\sum _{k=1}^{n}a_{i_{k}}v_{i_{k}}$ are meaningful. In summary style, we simply don't have the luxury to worry about such niceties as whether all bases have index sets/orders. Readers wanting a more leisurely definition can visit the main article (which is also wanting attention, by the way). Sławomir Biały (talk) 21:28, 27 August 2012 (UTC)

It is absolutely essential to not introduce side issues when making a definition, or else the definition will be wrong. (There is nothing preventing anyone from referring to some basis elements with subscripts later as needed. But it is important to make it clear that a basis is just a set of vectors, not a set of indexed vectors, and not an ordered set of vectors; those are additional structures that can be added later, but they are irrelevant to the definition of a basis.)

Someone's suggestion above of defining a basis as simply a linearly independent set of vectors that spans the vector space is excellent. That is the most direct and relevant way of formulating the definition. That is definitely a better suggestion than my item 4) at the top of this section.

I also believe that when there are extremely common logically equivalent definitions in mathematics, it is very important to mention them right away; in this case, a basis is a maximal set of vectors that is linearly independent, and a basis is a minimal set of vectors that spans the vector space, are the most natural two definitions that should accompany the original definition, as a theorem claiming that all three definitions are equivalent. Ideally in a form something like: "Let B be a subset of a vector space V. Then the following are equivalent: a) B is a basis; b) B is a maximal spanning set; c) B is a minimal linearly independent set."

Someone commented that "maximal" and "minimal" need clarification since they are vague. Well, maybe they ought to be clarified, but they are in no way vague; they are very basic technical terms in mathematics. A maximal set with property P is one that is not a proper subset of any set with property P. A minimal set with property Q is one that does not contain as a proper subset any set with property Q. That's all there is to it.Daqu (talk) 18:46, 28 August 2012 (UTC)

Now that the most important two-thirds of ideas of my controversial edit (hamhandedly reverted en masse) are back, and having had a few days to think about it, I think I'm willing to compromise the "ordered basis" issue with a footnote. It's not ideal, but students are going to encounter it this way in texts most of the time, and we can probably trust that they have more resources than WP to keep the distinction straight. I have come to believe that not saying to much is better, in this case. (As a sidenote, Daqu, you transposed maximal and minimal in your second to last paragraph, and you might want to switch 'em. I saw you wrote it correctly everywhere else, though!) Rschwieb (talk) 16:32, 30 August 2012 (UTC)

If we were writing a textbook, then I would agree that the "best" definition of a basis is as a spanning linearly independent set. However, in a textbook, there is ample time to dwell on things: giving examples, refining notation, etc. In an encyclopedia article, we just want to summarize the main points of various subsidiary articles (such as basis (linear algebra)). What appears in this article should be just the barest sketch of what appears there, including only an accessible overview, as well as any notation that is used in the rest of the article. We cannot dwell on irrelevant things like whether or not bases have index sets, are ordered, etc. Our readers do not expect a textbook treatment of such minutiae here, as they has nothing to do with the subject of this article. The subject of this article is "vector space", not "basis". If you have complaints about basis (linear algebra), then the correct forum for such a discussion is Talk:Basis (linear algebra), not here. Sławomir Biały (talk) 02:24, 1 September 2012 (UTC)

I agree that this is not the place to "dwell on irrelevant things". But it is essential not to even mention irrelevant things, particularly in introductory paragraphs.

There is no problem whatsoever with giving the basis vectors an index set -- best to call it something like J in keeping with tradition. (I was too hasty to say above that the basis vectors should not have subscripts.) But it is essential to say nothing that constrains the index set, since it could be any set, of any cardinality, and it could have nothing whatsoever to do with an "ordered" basis.

A footnote about ordered bases does not belong in introductory paragraphs. Since ordered bases almost always occur when the dimension is finite, it is fine to mention them in a section on finite-dimensional vector spaces -- that comes later. Regardless of the fact that students will encounter ordered bases, order plays no role in the definition of a basis and it is essential not to give the misimpression that it does. Or the misimpression that a basis can necessarily be indexed by some or all integers. (The usual Hilbert space -- a very important vector space -- having a countably infinite "Hilbert basis" is actually a vector space of uncountable dimension.Daqu (talk) 14:36, 3 October 2012 (UTC)

Since this discussion happened, I've seen that Basis_(linear_algebra) explains everything the way we would like to. I've been persuaded by the argument that vector space is not the correct article to be absolutely correct about bases, and that we should leave it to basis (linear algebra) is the right place. Rschwieb (talk) 16:00, 3 October 2012 (UTC)

If the concept of a basis were a complicated one, I might agree with you. Instead, it is exceedingly simple and central to any understanding of vector spaces. So there is no reason in the world that the mention of bases in this article should not be absolutely correct.Daqu (talk) 11:44, 4 October 2012 (UTC)

The reason given, that you are willing out of existence, is that it is a crucial fact about bases but not a crucial fact about vector spaces. There is no need for this article to duplicate the basis article in such detail. It would be best to say as little as possible here, and direct the reader to the basis article, which is much better suited to clarify the situation. Rschwieb (talk) 12:56, 4 October 2012 (UTC)

Concrete discussion of a few changes

1) "Spanning the whole space means" is clearer than "The former means". Arguments?

2) "Linear independence means that blah blah" is clearer than "Linear independence means that blah, (irrelevant digression into existence) blah". (Are we worried that when we say "This linear combination is unique" someone is going to interrupt us and ask "does that linear combination even exist?!".)

3) Part of the change I made (I haven't reintroduced it) before which didn't deal with index sets was to explain what we meant by minimality. Right now it says "linear indepenence=minimality=unique expression". I think it's pretty clear that we have failed to show what we really mean by minimality is: dependent vectors can be removed from the generating set without affecting the span. When you can't remove any more things without damaging the span, it's "minimal". I'll be waiting to hear everybody's comments for the next few days on these matters. Rschwieb (talk) 21:00, 27 August 2012 (UTC)

(1) agree. (2) agree. (3) agree to a point. I think we should state plainly, and in as few words as possible, what "minimality" actually means, and then get to the point about linear independence. I've gone ahead and tried to do this. Sławomir Biały (talk) 21:22, 27 August 2012 (UTC)

OK, glad to have the feedback. I'm happy with these changes. Rschwieb (talk) 16:27, 30 August 2012 (UTC)

OK, I also buy the "this is the wrong article to make the distinction" argument. Thanks for discussing it. Rschwieb (talk) 13:36, 4 September 2012 (UTC)

"Coordinatized viewpoint"

Sentence in section Bases and dimension
Since each vector in a vector space can be expressed uniquely by (1) as a linear combination of the basis vectors, and since the corresponding scalars ak can be viewed as generalizations of Cartesian coordinates, this point of view is referred to as the coordinatized viewpoint of vector spaces.

I believe that this sentence is irrelevant in a section that should be only a summary of a separate article (Basis (linear algebra), mentioned at the beginning of section as one of the main articles). Moreover, a search on Google shows that the expression "cooridinatized viewpoint" is never used elsewhere on the web, not even in the main Wikipedia article. The only websites which use this expression are plain copies of this article.

I propose to delete this sentence. Alternatively, it might be moved into the main article, provided that a reference is provided.
Paolo.dL (talk) 13:04, 31 August 2012 (UTC)

Surely the "whole point" of introducing bases in the first place is so there is some notion of coordinates. So I think your search methodology might be flawed. In any event, I have (essentially) rewritten the section. It was just too heavy-handed for a main article. It needs to present things in summary style. For precise definitions, the reader should consult the main articles on the respective topics (basis (linear algebra) and dimension of a vector space). If editors find these articles wanting, then surely they should improve them rather than whine that the top-level article vector space is incomplete! Sławomir Biały (talk) 00:14, 1 September 2012 (UTC)

Who is "whining that the top-level article is incomplete"?

This discussion is mainly about the expression "coordinatized viewpoint". As for the "notion of coordinates", my recent edit explains that the scalars a_k are called coordinates of v with respect to B. It might be also useful to mention that a_k can be viewed as a generalization of the Cartesian coordinates (e.g., x,y,z in R³). The fact that this is important enough to be mentioned in this summary seems to be just an opinion of yours, but I won't mind if this is mentioned.

On the contrary, the fact that this viewpoint is allegedly called the "coordinatized viewpoint" is irrelevant and not supported by references. Paolo.dL (talk) 12:46, 3 September 2012 (UTC)

" Who is "whining that the top-level article is incomplete"?" The previous thread, which was about this very same section. "The fact that this is important enough to be mentioned in this summary seems to be just an opinion of yours": Well, it's basically a centerpiece of any first course in linear algebra. See any linear algebra textbook. Sławomir Biały (talk) 13:25, 3 September 2012 (UTC)

OK, thanks for deleting the above mentioned sentence. See if you like my edit. Paolo.dL (talk) 11:27, 4 September 2012 (UTC)

Sense of a vector

Hello, I see that multiplying a vector by -1 yields to a vector pointing in "the opposite direction". Shouldn't it be the "opposite sense"? In fact, shouldn't we have a sense (vector) article redirecting there please? We need it at eigenvector. Thanks. 219.78.115.252 (talk) 14:13, 14 October 2012 (UTC)

While I understand that some people use "sense" this way, "direction" seems to be much more common, and I have typically only heard people use "direction" in practice. I don't see the need for an entire article for a synonym of direction (geometry). If you've got good ideas for additions, it seems like they'd be most useful at Direction (geometry). Rschwieb (talk) 12:58, 15 October 2012 (UTC)