Talk:Vector space

Merge with Vector (mathematics and physics)
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, 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, which redirects presently to Euclidean space. The first article could be named , (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, 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)

I would consider vectors as $$\mathbb{R}^n$$ as being very distinct from vectors as $$(G, +, \cdot) \text{ where } (\cdot) : \mathbb{R} \times G \to G \text{ and } (+) : G \times G \to G$$. After all, we don't equate Binary tree and Magma (algebra). Further, $$\mathbb{R}^3$$ also has $$(\times)$$. 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 : 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)


 * I also disagree with the merge, and I'll be removing the templates. A vector isn't a vector space, and the properties of vector spaces are vastly different than what vectors themselves are. &#32; Headbomb {t · c · p · b} 14:28, 24 March 2020 (UTC)

Merge list of related vector concepts
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
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: 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)
 * 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.
 * D.Lazard (talk) 10:08, 15 February 2022 (UTC)

"Abstract vector space" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Abstract_vector_space&redirect=no 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  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)
 * 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)
 * 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)

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)