Talk:Vector space
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Merge with Vector (mathematics and physics)
[edit]- 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.
- Closed as there is no consensus to merge Vector (mathematics and physics) into Vector space Felix QW (talk) 12:11, 14 January 2022 (UTC)
Hi, It seems to me that this page covers the same topic as Vector (mathematics and physics), just from a slightly different point of view. I will therefore suggest merging that page into this one, to avoid having a redundant page, and to have one place for people to look instead of two.
- Ramzuiv (talk) 22:50, 5 February 2020 (UTC)
- The main motivation for having two article is the sentence
Therefore, one talks often of vectors without specifying the vector space to which they belong.
, in the lead of Vector (mathematics and physics). In particular, many people know of displacement vectors, position vectors and velocity vectors without knowing any mathematical definition of the vector space to which they belong. So, there is a need of two different articles. However, Vector (mathematics and physics) would deserve a major edit for being better adapted to this sort of readers. D.Lazard (talk) 09:13, 6 February 2020 (UTC)
- There is also Euclidean vector, which is the target of the redirect
vector (mathematics and physics)vector (physics). --JBL (talk) 12:28, 6 February 2020 (UTC)- Presently none of these articles is a redirect. I suggest to merge them into the second one, and to redirect the first one either to the merged article or to Euclidean vector space, or, more exactly to the target of this redirect. D.Lazard (talk) 12:58, 6 February 2020 (UTC)
- Sorry, I have corrected my error, I meant vector (physics). --JBL (talk) 13:01, 6 February 2020 (UTC)
- There's certainly more pages about vectors than there should be, I'd like to see something merged into something. I'd suggest combining Vector (mathematics and physics) and Vector space to be one page that introduces vectors as a general concept, and keep Euclidean vector as discussing a specific type of Vector. I really don't see the need for two different pages that describe the general concept of vectors from two slightly different angles. It does seem to me that it's worth keeping a distinction between Euclidean Vectors and Euclidean Space - Ramzuiv (talk) 00:58, 12 February 2020 (UTC)
- I agree that only two (or maybe three) article are needed. One about vectors that are considered independently of the structure to which they belong, and the other, vector space, about the structure. The third possible article could be Euclidean vector space, which redirects presently to Euclidean space. The first article could be named vector (geometry), (presently a redirect to Euclidean vector), as it concerns usages relative to classical (Euclidean) geometry. This includes usage in physics, and specially in mechanics. Presently, Euclidean vector space is defined as a part of the modern definition of Euclidean space. Expanding it as a stand alone article could allow explaining more clearly the relationship between the content of the two other articles.
- IMO, such a major change of the organization of our articles must be prepared by writing first drafts of the future articles, and then having a wide discussion on explicit projects. Are you willing to write these drafts? D.Lazard (talk) 10:28, 12 February 2020 (UTC)
- There's certainly more pages about vectors than there should be, I'd like to see something merged into something. I'd suggest combining Vector (mathematics and physics) and Vector space to be one page that introduces vectors as a general concept, and keep Euclidean vector as discussing a specific type of Vector. I really don't see the need for two different pages that describe the general concept of vectors from two slightly different angles. It does seem to me that it's worth keeping a distinction between Euclidean Vectors and Euclidean Space - Ramzuiv (talk) 00:58, 12 February 2020 (UTC)
- Sorry, I have corrected my error, I meant vector (physics). --JBL (talk) 13:01, 6 February 2020 (UTC)
- Presently none of these articles is a redirect. I suggest to merge them into the second one, and to redirect the first one either to the merged article or to Euclidean vector space, or, more exactly to the target of this redirect. D.Lazard (talk) 12:58, 6 February 2020 (UTC)
- There is also Euclidean vector, which is the target of the redirect
I would consider vectors as as being very distinct from vectors as . After all, we don't equate Binary tree and Magma (algebra). Further, also has . Theanswertolifetheuniverseandeverything (talk) 13:56, 26 February 2020 (UTC)
- I disagree merging this article with Vector (mathematics and physics). Instead, the latter article should be improved by making it into more than a kind of verbose disambiguation page. This article here is about the concept in linear algebra and its connections to other (mathematical) fields. Also, from a practical point of view, I agree with D.Lazard: if we were to merge the articles, it would be an enormous amount of work to get the result to a quality which is near the one of this article (right now). Unless someone is committed to such a task very definitely, both articles are likely to be a mess afterwards. I really think improving the Vector (mathematics and physics) is the thing to be done here. Jakob.scholbach (talk) 15:19, 27 February 2020 (UTC)
Merge list of related vector concepts
[edit]- 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)
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)
"Generalizations" section
[edit]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)
"Abstract vector space" listed at Redirects for discussion
[edit]
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
[edit]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
[edit]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
[edit]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
[edit]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.
- Article (edit | visual edit | history) · Article talk (edit | history) · Watch • • 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
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Some works
[edit]- 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)
- @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)
- 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)
- 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)
- 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)
Section 2
[edit]@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)
- @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)
- 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)
Presentation of the Definition
[edit]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)
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