Talk:Vector space

Merge list of related vector concepts

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

No opposition to section move. Felix QW (talk) 19:17, 10 May 2022 (UTC)

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Vector (mathematics and physics) is being setup in summary style, so that broad-concept article should not introduce anything that is absent here. fgnievinski (talk) 05:52, 1 November 2021 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

"Generalizations" section

As I understand the word "generalization" in math, one says that objects of type A generalize objects of type B if certain A-objects are B-objects. So:

• it is perfectly correct to say that modules generalize vector spaces.
• Affine spaces do not generalize vector spaces; any vector space defines an affine space, but it is not the case that certain affine spaces are vector spaces.
• One could consider those certain vector bundles in which the base space is a point (I note this is not even mentioned in the article), and to then identify the total space with a vector space. It may be overly pedantic to say that the extra specification of the particular point matters, but I think it is simply true. It may be more correct to say that "vector space" is generalized by the concept of total space of a vector bundle, and not by vector bundle itself.

Sorry for the pedantry, but the article seems to suggest that all three concepts are equally well generalizations of vector space, and maybe this is not so good. Gumshoe2 (talk) 07:23, 15 February 2022 (UTC)

 Fixed D.Lazard (talk) 10:08, 15 February 2022 (UTC)

"Abstract vector space" listed at Redirects for discussion

The redirect Abstract vector space has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 June 24 § Abstract vector space until a consensus is reached. Hildeoc (talk) 00:59, 24 June 2023 (UTC)

Counterexamples of mathematical structures for each axiom failing while meeting all the remaining ones

They are supposed to be axioms. Given the amount of time vector spaces have been defined, I would assume there would be a counterexample showing some mathematical object satisfying all axioms except one.

Counterexample help understanding just as much as examples. Sometimes more. It certainly would be helpful for me. Ndhananj (talk) 22:57, 20 August 2023 (UTC)

Vector addition and other operations in vector component format

The page would gain comprehension and usefulness if a sub-section in the respective operations sections is also devoted to using the operation using component vectors. For example, in the addition of two vectors, using |R| = √((Ax + Bx)² + (Ay + By)²) and Angle between the resultant and base vector = tan^(-1) ((Ay + By)/(Ax + Bx)). R0ck$ (talk) 05:46, 30 October 2023 (UTC)

@R0ck$: I'm not sure what you mean. This is an article about general vector spaces that might have no inner product associated with them, and which might be multidimensional. There is an example (ordered pairs of numbers) that showcases components, but in no way are the definitions dependent on the existence of a basis.--Jasper Deng (talk) 06:11, 30 October 2023 (UTC)

@Jasper Deng apologies, I am not well-versed at all about "inner product" as a terminology. Will educate myself about the same and try to understand what you are trying to say.

All I was concerned about was that frequently it is useful to perform a vector sum in a manner that separates the magnitude and direction, and the formula for the same does not seem to be there in this particular article. R0ck$ (talk) 11:01, 30 October 2023 (UTC)

@R0ck$: I don't think you understand. "Magnitude and direction" are only meaningful in inner product spaces, like for Euclidean vectors. Adding and subtracting vectors is always possible using components in a basis, but it is nontrivial to show that a basis always exists and certainly not part of the definition. For many infinite-dimensional vector spaces, you are not going to be adding componentwise: the set of all real-valued functions on a given set forms a vector space, but any basis is going to be of uncountably large dimension. If anything, the magnitude and direction formulation is emphatically not what this article is getting at.--Jasper Deng (talk) 11:03, 30 October 2023 (UTC)

Scalar multiplication is not a binary operation

The definition erroneously said there are two binary operations, one of which is scalar multiplication. In fact, the group of units of the field produces a group action on the vectors. Inclusion of zero for scalar multiplication annihilates the vector space to the zero vector. Binary operations require one set, but scalar multiplication starts with two: F and V. Precision is mathematics is expected, and this hitch of imprecision might have led to confusion. Reference has been made to binary function since introduction of group action at this level assumes much of the reader. — Rgdboer (talk) 02:41, 22 December 2023 (UTC)

Please, read the third paragraph of Binary operation. If you find it erroneous, you must discuss it on that talk page. In any case, most textbooks call scalar multiplication a binary operation. 09:49, 22 December 2023 (UTC) D.Lazard (talk) 09:49, 22 December 2023 (UTC)

That paragraph in Binary operation conflicts with the definition given in the article proper. In usage it seems Scalar multiplication is a legacy exception, or carveout, as the usage perpetuates a misnomer. The paragraph seems appropriate, given usage, particularly as Binary function is mentioned as a valid alternative. — Rgdboer (talk) 00:57, 27 December 2023 (UTC)

As far as I know, most textbooks on vector spaces use "binary operation" for the sclar multiplication, and not "binary function". So, Wikipedia must follow the common usage. D.Lazard (talk) 09:01, 27 December 2023 (UTC)

Composition is the word used by Emil Artin in his Geometric Algebra (book), available via page 4, Internet Archive. — Rgdboer (talk) 00:56, 4 January 2024 (UTC)

GA Reassessment

Vector space

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Article (edit | visual edit | history) · Article talk (edit | history) · Watch • Watch article reassessment page • Most recent review

Result: Kept. ~~ AirshipJungleman29 (talk) 19:02, 13 February 2024 (UTC)

This 2008 listing contains significant amounts of uncited material, far beyond what WP:CALC permits, and thus does not meet GA criterion 2b). ~~ AirshipJungleman29 (talk) 03:00, 25 January 2024 (UTC)

Pinging @Jakob.scholbach and @Ozob who nominated and reviewed the GA for the first time, respectively. Dedhert.Jr (talk) 12:24, 25 January 2024 (UTC)

FYI I notified both on their talk already. ~~ AirshipJungleman29 (talk) 12:45, 25 January 2024 (UTC)

This is all easily verifiable basic material. @AirshipJungleman29 why don't you try to add some sources instead of spending all of your time demanding that other people jump through made up hoops. –jacobolus (t) 18:05, 25 January 2024 (UTC)

On the worth of the GAR process and participating in it

You are free to jump, or not jump, jacobolus. You have spoken on the worth of the GA process ("Arguing about whether it ticks off some boxes on a made up checklist (a poor proxy for article quality) is a total waste of time.") and on your disinclination to engage with any part of it ("Let the bureaucrats take away the little green badge if they must"), so I really cannot see why you should care if other people decide to jump when I ask. Some have, many won't—life goes on. ~~ AirshipJungleman29 (talk) 18:33, 25 January 2024 (UTC) This article is objectively "good" compared to the typical for Wikipedia articles. Instead of adding a bunch of perfunctory footnotes to arbitrarily chosen introductory textbooks that any curious reader can find for themself with about 1 minute of effort, it would be much more useful for you to find a currently mediocre article and work to get it to higher quality. The way you're doing this now is inducing a bunch of experienced and careful editors to spend a bunch of work on frankly marginal activities that are a relative waste of time; you and they would be doing more good for the Wikipedia project if they picked something (just about anything) else to work on. As another example, it looks like this kind of thing went a substantial distance toward exhausting User:XOR'easter's motivation; that alone is enough damage to more than counterbalance any good that will come of this whole exercise. –jacobolus (t) 18:54, 25 January 2024 (UTC) Thanks for telling me what I like to do with my free time. As it happens, I quite like doing this. I apologise if you find that objectionable jacobolus, but the consensus of the Wikipedia community is that it is a useful activity. If you feel that is not the consensus, you are free to propose deprecating the GA process at WP:VPR. Incidentally, XOR was not the only one impacted by that GAR; its proposer was also considerably jaded, and stepped away from the site for a few months. I suspect it was more the peculiarities that discussion, rather than the process as a whole, which caused the casualties. I think we can agree in hoping that sort of thing won't happen again. ~~ AirshipJungleman29 (talk) 19:08, 25 January 2024 (UTC) The problem is not what you are doing with your free time. The problem is that you are imposing unreasonable demands on other editors, asking them to devote a considerable amount of their own time and effort to cleaning up articles that should not be prioritized. If it is so important to you that these articles get cleaned up, then put some skin into the game. Put a few hours or weeks into sourcing each article yourself, before dragging it to GAR. —David Eppstein (talk) 19:22, 27 January 2024 (UTC) David Eppstein, I prioritize these articles because they have this little green blob that Wikipedia have decided means the article meets a certain set of criteria, and thus I think (perhaps wrongly) that they should meet said criteria. If you are not willing to devote time and effort to making the articles meet the criteria, because you feel they should not be prioritized, simply let it be delisted. If you feel you don't currently have the time to "jump through made up hoops"—perfectly fair, you have done so 115 times already at WP:GAN—you can simply do so later and renominate it there. I hope you understand my reasoning. ~~ AirshipJungleman29 (talk) 15:55, 28 January 2024 (UTC) You are entirely welcome to devote your own time to improving GA articles to meet your standards of what GA articles should be. However, what I see you doing instead is making work for other people by nominating article after article for review, without any evidence of putting effort into cleaning up those articles yourself first. It comes across as selfish and thoughtless. And the net effect has already been to drive authors away from the GAR process, as they were previously driven away from the FAR process, because rather than being something that one can do and move on, it turns into a never-ending time sink of pointless re-reviews. —David Eppstein (talk) 20:01, 28 January 2024 (UTC)

Some works

• I'll do the work in the section on "Related topics and properties". It has some awkward list that can still be improved. Such works are in my sandbox. Dedhert.Jr (talk) 02:39, 26 January 2024 (UTC)

• Done. But need more sources and some copyedits. Dedhert.Jr (talk) 03:15, 26 January 2024 (UTC)

• I can provide sources for whatever specifically needs citations, but I don't currently see things where I would want to have an additional source. If you want, add a citation needed tag and I'll take care. Jakob.scholbach (talk) 22:27, 4 February 2024 (UTC)

• The nominator Jakob.scholbach has signalled their intention to work on the article on my talk page. As usual, this reassessment should be left open as long as they intend to work on it, up to a maximum of around three months. ~~ AirshipJungleman29 (talk) 18:21, 28 January 2024 (UTC)

  @AirshipJungleman29, @Jakob.scholbach. I have seen your conversation about the improvement of the article on the talk page. Since the main author is busy in real life, I think I can take over temporarily by adding some sources and copyedits to this article. However, some comments may require some clarification from the main author directly. Dedhert.Jr (talk) 02:05, 31 January 2024 (UTC)

  Thank you! Jakob.scholbach (talk) Jakob.scholbach (talk) 19:37, 4 February 2024 (UTC)

• Some see-below links are tagged with clarity. I cannot find where the see-below links are redirected to. Dedhert.Jr (talk) 02:02, 31 January 2024 (UTC)

  I have removed these, they didn't really add any value (even if they would have worked). Jakob.scholbach (talk) 19:42, 4 February 2024 (UTC)

• Are the types of vectors in the section Vector space § Related concepts lists merely? I do think these should be removed, IMO Dedhert.Jr (talk) 02:32, 31 January 2024 (UTC)

  I agree! Jakob.scholbach (talk) 19:37, 4 February 2024 (UTC)

• Do we have to write the homogeneous differential solutions in a "linear equation" section? It seems more difficult to comprehend, and are those related? Dedhert.Jr (talk) 03:49, 2 February 2024 (UTC)

  Yes they are closely related. The point is that the process of taking a function f and associating to it its derivative (or second derivative etc.) is linear as well, which is why the solution spaces of homogeneous differential equations are vector spaces.

  I think the discussion there takes so little space that this seems OK to me. Jakob.scholbach (talk) 19:39, 4 February 2024 (UTC)

• AirshipJungleman29 : please give an update on where specifically you think citations are needed. (I didn't check how many / which ones have been added very recently by Dedhert.Jr and maybe others). The article currently has, IMO, a fair amount of citations overall, and it would be pointless to just add 20 more on generic grounds. Jakob.scholbach (talk) 22:27, 4 February 2024 (UTC)
  • Hi Jakob.scholbach, the GA criterion 2b) has been modified so that all content that could be challenged and doesn't fall under WP:BLUESKY needs to be cited inline. See e.g. Descartes' theorem, currently at GAN, for something that does this well. I understand that you could see this as tiresome and/or pointless, but that is what the GA criteria ask for, and it is a lot easier than some articles which come to GAR needing to be entirely rewritten. ~~ AirshipJungleman29 (talk) 12:46, 5 February 2024 (UTC)

    @AirshipJungleman29 Correct me if I'm wrong. The GACR2b has been modified, stating that the article has no original research but rather covered with the verifiability in the reliable sources and citation inlines, with the exception that plot summary or explanation do not need to be sourced. However, some of the paragraphs are not plot summaries, or somewhat backgrounds to describe the highly technical topics. Should I added the citation-tag in this case? Dedhert.Jr (talk) 08:32, 6 February 2024 (UTC)

    OK, AirshipJungleman29 so I reiterate my request to please name a few specific instances of claims / statements you think require additional sources. What is the content that could reasonably be challenged? Jakob.scholbach (talk) 09:06, 6 February 2024 (UTC)

    Jakob.scholbach, thanks very much for your work on the article so far. I have tagged a few places where inline citations would be helpful; please let me know if you think any of them fall under WP:CALC. ~~ AirshipJungleman29 (talk) 22:38, 8 February 2024 (UTC)

    I have removed to citation needed tags: in one case I decided to delete the paragraph containing it, because it was out of place there, in the other case (about addition of complex numbers) it was falling under WP:BLUESKY.

    For the other tags, I did supply references. Let me, however, state quite clearly that this kind of citation needed request is hardly a service to anyone on Wikipedia: it was in these cases a trivial matter to find the required assertion in the subarticles, or to pull up various sources at once. Notice how the references are often to the very first pages of some book, highlighting how strongly these assertions fall under the rubric "not-challengeable".

    AirshipJungleman29, with all respect to your principles about your work on GAN, I can't refrain from reiterating comments made by jacobolus and David Eppstein: I suggest we all spend our time on better things. Jakob.scholbach (talk) 14:14, 10 February 2024 (UTC)

    Thank you very much for your work Jakob.scholbach; for myself, I will continue to work at GAR until the community decides to deprecate the process. It is not all I do on Wikipedia—see today's featured article on the main page—but I find this to be worthwhile in itself. You are welcome to decide whether you have better things to do than provide trivial citations in the future. Thank you also for your cordiality in your responses. ~~ AirshipJungleman29 (talk) 19:57, 10 February 2024 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Section 2

@Dedhert.Jr Thank you very much for your efforts on this article! What exactly though is the rationale behind the rearrangement of the second section? It seems to me to be a lot more difficult to understand in the new, compressed form, and for instance the definition of linear independence as "the linear combination that is equal to zero" has little resemblance to the usual definition, according to which a set of vectors is linearly independent if there is no non-trivial linear combination of those vectors that equal 0. Felix QW (talk) 19:03, 3 February 2024 (UTC)

@Felix QW My opinion about rearrangement is that there are some relation between basis and linear combination, and this could be explained in one single paragraph rather than described in list. Dedhert.Jr (talk) 06:16, 4 February 2024 (UTC)

I partially reverted the change for now, as I think it is important to have correct and clear definitions of the basic concepts in our vector space article. While I think the structured pairs of concept and definition work well, I would be fine with any other layout, as long as the definitions themselves are preserved. Felix QW (talk) 08:28, 4 February 2024 (UTC)

@Felix QW You partially reverted the edit, but it could also mean that you deleted more citations to be added. Can you please add them up? Dedhert.Jr (talk) 09:25, 4 February 2024 (UTC)

@Felix QW Nevermind. I will added it later. Dedhert.Jr (talk) 09:38, 4 February 2024 (UTC)

Presentation of the Definition

The presentation of the definition is not very good. A clear definition should IMHO mention all the components used. Furthermore, writing scalar multiplication not explicitly might be okay when working with vector spaces daily, but a definition should make this explicit.

Also in the table of "axiom" the header "meaning" is a bit misleading, like that there is some room for interpretation, but actually whats given in the column is the definition. 132.176.73.161 (talk) 06:56, 19 April 2024 (UTC)

I do not understand your second concern, since the word "meaning' does not appear in the article. I do not understand either your first concern. By "component", I suppose that you mean coordinates on a basis. This cannnot appear in the definition since not all bases are finite, and many vector spaces do not have a given basis. For example, the real valued functions with the reals as domain form a vector space for which no basis can be explicitly described. D.Lazard (talk) 13:20, 19 April 2024 (UTC)

Removal of 1*v = v as an axiom

I suggest removing 1*v = v as a defining axiom of a vector space and instead show that it is a property that follows from the other axioms. Note that once we have shown that 0*v = 0 (zero vector) and -v = (-1)*v, both of which follow from the other axioms, we have that:

0 = (1 + (-1))*v = 1*v + (-1)*v <=> 0 + v = 1*v + -v + v <=> v = 1*v. 68.193.34.74 (talk) 11:09, 2 October 2024 (UTC)

You wrote "-v = (-1)*v, which [...] follows from the other axioms". This is true, but this cannot be used to prove 1*v = v, since 1*v = v is used for proving -v = (-1)*v (circular reasoning). D.Lazard (talk) 14:11, 2 October 2024 (UTC)

Good point. However, after further consideration, I still conclude we do not need to assume 1•v=v as an axiom of a vector space but rather can prove it from the other axioms, alone.

By the axiom of the compatability of scalar and field multiplication in vector spaces, we have that a•(b•v) = (ab)•v, for all a and b in our base field and all v in V (a vector space). To be clear, I am using • to symbolize scalar multiplication (i.e., • : F x V --> V) in an F-vector space, and juxtaposition to indicate field multiplication (i.e., F x F --> F) in the base field F.

Since the above axiom holds for all values a in F, set a = 1, i.e., the number 1 (multiplicative identity) from our base field F. Then:

1•(c•v) = (1c)•v (axiom of compatability of scalar product and field multiplication)

= c•v (multiplicative identity axiom, i.e., 1c=c for all c in our base field F)

By the axiom of closure of scalar product in V, c•v is an element of V for all c in F and all v in V, so we may define x := c•v where x is an element in V. Then, substituting above and reading the extreme left and right hand sides, we have 1•x = x.

I checked carefully but did not find any circular reasoning problems with my argument. In particular, the axiom of the existence of a multiplicative identity 1 in F such that 1c=c for all c in F is part of our assumption that V is an F-vector space over F, a field.

What am I missing? 68.193.34.74 (talk) 13:45, 11 November 2024 (UTC)

You prove that 1•x = x for every x such that there exist c and v such that x = c•v, but you do not prove that every element of the vector space can be factored this way. D.Lazard (talk) 22:10, 11 November 2024 (UTC)

If you consider the scalar multiplication c•v = 0 for every c and every v, then every abelian group satisfies all axioms of a vector space except that 1•v = 0 ≠ v. This shows that the axiom 1•v = v is really needed. D.Lazard (talk) 22:48, 11 November 2024 (UTC)

Thanks for your response. I fully agree that 1•v = v is indeed necessary for the reasons you indicated and that it cannot be proven from the other axioms.

After spending (way too much) time thinking about this, I realized many other problems arise if we do not assume 1•v = v.

First, with your proposed definition of scalar multiplication sending everything to the zero vector, there is no field F that is a vector space over itself. In particular, if • : F x F --> F is defined as you indicated, then for all field elements a, b (both nonzero), we have a•b = 0 = 0, meaning that a and b are zero divisors in F, a contradiction because F is a field (hence an integral domain, hence has no zero divisors). I guess this argument assumes that scalar multiplication IS field multiplication, but I think that is the usual assumption when considering F as a vector space over itself.

Another thing I noticed is that the operation of • "looks like" a left group action, albeit of F* (nonzero field elements), viewed as a group under field multiplication with identity 1, on V. Moreover, the distributivity axioms of a vector space further imply that 0•v = 0 so that • is also a well-defined map of all of F (and not just F*) on V. However, we no longer have a group action if 1•s = s is not assumed. This seems to cause all sorts of problems I had not even considered. Among other things, a group action of G on another group H is supposed to permute the elements of H, which we do not get if everything is mapped to 0.

I am curious if it is indeed possible to reconstruct vector spaces by starting with V as an abelian group and then defining the scalar product as being a left action along the lines above. In any event thank you for disabusing me of my idea to discard 1•v = v as an axiom! 68.193.34.74 (talk) 14:31, 20 November 2024 (UTC)