Talk:Vector space/Archive 2

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Include this article in Vector Calculus

Is it OK to include this article in the Vector Calculus category? And if not, please point out why. It helped me to see the vector space after I read about the vector field. —The preceding unsigned comment was added by Muttley.meen (talk • contribs) 05:49, 16 May 2007 (UTC).

Relevance of History of vector spaces

Why is Hausdorff mentioned in the article?—Preceding unsigned comment added by 68.148.232.162 (talk • contribs)

linear independence

Would any mathematician be kind enough to help me to know if the square roots of primes in the vector spce of real numbers over the field of rational numbers(with usual + & .) are linearly independent? if so, please prove it for me  : Sumita Garai—Preceding unsigned comment added by Sumitagarai (talk • contribs)

help needed

I tried to add a reference to the equivalence result for Choice and vector space bases. I can't get it to work. Could somebody assist me and explain to me what I did wrong in the source? Thanks in advance. —Preceding unsigned comment added by 68.40.81.16 (talk) 00:04, 6 April 2008 (UTC)

[1] I created a new section Notes where the footnote now appears, using the <references/> syntax. --Newbyguesses (talk) 00:21, 6 April 2008 (UTC)

Thanks! I will try to correct the naming error (it shows "Andreas" instead of "Andreas Blass" at the moment). Didn't know that much about Notes and references system. Thank you for explaining it. —Preceding unsigned comment added by 68.40.81.16 (talk) 04:12, 6 April 2008 (UTC)

I have given up for now to try to get a perfect result on the naming problem. Is it a bug in the citation templates? I tried to follow Wikipedia:Citation_templates. cite conference truncated the last name adding it to the authorlink instead. With Citation it's better, but still a pipe symbol within the links on the actual wikipage, the link itself is correct -- which I considered more important for now. —Preceding unsigned comment added by 68.40.81.16 (talk) 04:24, 6 April 2008 (UTC)

I tried to fix the reference for you. Did you look at Template:Citation? That explains how to use the citation template. -- Jitse Niesen (talk) 12:57, 6 April 2008 (UTC)

Bad Pun

"More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms which are listed below."

Well, of course a vector will follow an axiom. An axis is typically represented as a vector, is it not? --74.195.97.48 (talk) 03:13, 7 July 2008 (UTC)

Rewrite advisable

This article is too focused on a set of limited ideas about vector spaces. It should provide a general introduction at the outset to the topics simply linked below, for example, in the subsections on topological spaces (namely Banach spaces, inner product spaces, function spaces etc.). This overall article should lead the reader through this morass of other articles on the subject, which would be more illuminating and more useful than an attempt to become yet another bead in a long string. Brews ohare (talk) 16:45, 14 September 2008 (UTC)

Absolutely. I'm just about to start this endeavour. Are you in? Jakob.scholbach (talk) 16:49, 14 September 2008 (UTC)

I am happy to help, but frankly insufficiently educated to carry out the full program. Brews ohare (talk) 17:10, 14 September 2008 (UTC)

Reversing this direction of thought,

A paragraph on Hilbert space begins with this phrase, but apparently the "line of thought" being reversed got dropped, leaving the reader wondering what line of thought that was anyway. Brews ohare (talk) 17:18, 14 October 2008 (UTC)

Yes, that was due to a restructuring of mine. I will brush over the text anyway, once a reasonable amount of content is achieved. Especially applications is still lacking. Any ideas? The problem is a bit, many many things in math are vector spaces, but we ought to find applications that are somehow using the vector space structure in a particular way. Jakob.scholbach (talk) 21:01, 14 October 2008 (UTC)

Notes and citations

I don't see the benefit of separate sections for notes and "citations". There isn't much difference between "See also Jordan canonical form and Roman 2005, ch. 8, p. 140." and "Treves 1967, Theorem 11.2, p. 102." The purpose of such a scheme is to separate citations (i.e., sources) from explanatory footnotes. That isn't necessary here, as the sources are in a separate "references" section anyway. An appendix structure like this is nonstandard per WP:LAYOUT, so I have merged it into a standard format. Many of the explanatory notes have or need sources anyway. Geometry guy 19:31, 18 October 2008 (UTC)

Frankly, I'm not entirely happy with this step. (Aside: I'd personally prefer one section with all extra stuff (citations, footnotes, references, see also, external links), simply to avoid cluttering up the TOC). Separating the two entities seems a plus to me, especially when we are adding more and more refs (compare with groups, e.g. so we will easily have 60+ reference-style-footnotes and perhaps 20-30 content-footnotes). In this context, separating the footnotes which give additional content information (such as "this is a conseq. of Hahn-Banach...") seems clearer from an organizational point of view. Obviously, tweaking one or another footnotes, such as "See also Jordan canonical form and Roman 2005, ch. 8, p. 140." (which should actually go to the main text)" is certainly in order. Jakob.scholbach (talk) 08:39, 20 October 2008 (UTC)

I agree with Jakob. I am one of those people who always read all footnotes to see if there is any interesting additional information there. Therefore I hate books with endnotes, especially if most of them only contain boring references but a few have interesting text. When I check a Wikipedia maths article, I have to click on all footnotes, and more often than not I am frustrated because they don't say anything interesting (just "author, year" for a statement which I think doesn't need a citation anyway because every mathematician knows it). It was a pleasant surprise when I saw the elegant solution at group (mathematics), and I would like to see this at all longer articles. --Hans Adler (talk) 09:34, 20 October 2008 (UTC)

Since this style passed FAC without comment, I'm not going to dig my heels in. However, distinctions are easily blurred here: e.g., at Group (mathematics), footnote 2 contains a valuable quote, footnote 37 contains further information and wikilinks, and footnote 60 contains useful specific advice to the reader. By contrast footnote c is a reference, footnotes e, g, o, p, v, x and y are unsourced, and footnotes i, j, v, x and y are not much more than wikilinks. Several of these and other footnotes should really be in the body of the article: footnotes can encourage lazy writing, as KSmrq used to point out.

All footnotes lead the reader to additional information, whether they be wise words, wikilinks, or links to sources. The solution to the problem of bland references with page numbers is not to place them in a ghettho, but to enhance their value. Here I've learnt from some younger editors recently: see e.g. Endomembrane system. Increasingly, sources have online versions, and it is possible to link specific chapters directly, and even highlight search terms. This blurs the distinction further between footnotes with wikilinks and footnotes with weblinks. Geometry guy 20:46, 20 October 2008 (UTC)

Hm. The majority of sources are books, which are hardly ever fully accessible online. Google books sometimes has a few pages, but I'm reluctant linking to a page that displays only two or three pages in a row. I guess I will leave the formatting as it is now, but I'm fairly convinced that putting everything into one section is more ghettoish than having a section on "book, chapter, page"-style notes and a section with additional explanatory notes. Considering the latter on a case-by-case-basis doesn't preclude putting footnotes, IMO. We can decide on what formatting is best when the article has reached some stable level.Jakob.scholbach (talk) 08:58, 21 October 2008 (UTC)

I changed it back. IMO, it's just better if you read the text and have a footnote "nb 1", which tells you that there is more than just "Th. x.y., pp. z", so it's worth scrolling down. Jakob.scholbach (talk) 19:42, 7 November 2008 (UTC)

Replacement for Image:Linearsubspace.svg?

I uploaded a different version (Image:Linear subspaces with shading.svg). Should I change the article to use it? Or is the colour scheme of the new file too yucky? (The purple is a bit overpowering...)

Alksentrs (talk) 01:28, 19 October 2008 (UTC)

Great: the image is much clearer! The purple is yucky, I agree. Can you change it easily? See e.g. Image:Apollonius8ColorMultiplyV2.svg for colour scheme ideas. Geometry guy 09:27, 19 October 2008 (UTC)

I've uploaded a new version, with a grey ground. Alksentrs (talk) 10:33, 19 October 2008 (UTC)

Thanks. (just an aside to Alksentrs: for accessibility reasons, it is worth thinking about colors that are well distinguishable--check out User:KSmrq's list of supposedly "real" opposite colors. the colors you used seem pretty close to the one I had earlier (which were from Ksmrq's list)).

Another image-related issue: do you know what to do about Image:Moebiusstrip.png? A while ago, I uploaded a new version, with different colors, so that the strip is blue, and not (as still shown in the article, notice also that it is shown green in the thumbnail list at http://en.wikipedia.org/wiki/Image:Moebiusstrip.png) green. Weirdly, if you click on the image, it does display the new version of the file. Does WP have an image cache? How could one purge it? Jakob.scholbach (talk) 08:31, 20 October 2008 (UTC)

I think it does have a cache of rasterized SVGs of various sizes - I just changed the size by one pixel and now the article shows the blue version... the image's history still shows green though... Alksentrs (talk) 09:45, 20 October 2008 (UTC)

Order of the article or "why are bases introduced so late"?

Why are bases and dimension introduced so late in the article? At the moment bases are introduced after matrices, eigenbases(!), tensor product, etc. The discussion of all these topics could benefit from actually having the concepts of basis and dimension to work with. The question then is where to put this section. It certainly must come before the section discussing linear maps, but I'm somewhat in dubio about whether to put it before or after the examples section. Any toughts? (TimothyRias (talk) 08:46, 21 October 2008 (UTC))

You are perfectly right, that the presentation of the whole article still has to be reworked. For example, eigenbases without bases is clearly a misconstruction. For the moment I focus(sed) on putting the content we will eventually need. Rounding it off will be the next step.

However, I disagree that bases "certainly must come before linear maps". It is a general principle that objects are best understood by morphisms, i.e. here linear maps, between them. For example, the category consisting of non-negative numbers, together with matrices of the appropriate rank, is an equivalent category. So, it is important to understand maps. It is also, as a matter of fact, a lot easier to understand a linear map than what a basis is. So I personally prefer this order. Thirdly, many readers won't want to know to much about bases, but will be interested in the relation to linear equations and, thus, linear maps. Jakob.scholbach (talk) 11:58, 21 October 2008 (UTC)

Well I for see some issues, if linear maps are discussed before bases and dimension. For example, representing linear maps by matrices uses bases at a conceptually crucial point. It would also be nice to be able to say that the dimensions of the matrix will match the dimensions of the vector spaces etcetera. I also disagree that it is a "matter of fact" that linear maps are easier to under stand then bases. Some of the subtle points about bases of infinite dimensional vector spaces maybe difficult, but the concept that a (finite dimensional) vector space is spanned by a certain number of base elements (with the number being its dimension) is in fact very intuitive. Abstract linear maps (certainly for infinite dimensional spaces) are actual conceptually quite hard for somebody not familiar with the concept of a map. First discussing bases opens the way for first discussing matrices and giving linear maps as the technical generalization. Matrices are part the high school curriculum in many countries and should thus be familiar to most interested readers. Also, you don't actual really need linear maps to understand relation between linear equations and vector spaces.

As note on the side the current section on bases is very technical. If we decide to move it up it should probably be rewritten in a more accessible manner. I'll have a go at this if we decide to go in that direction. (TimothyRias (talk) 14:15, 21 October 2008 (UTC))

You have some good points. The point in telling what a linear map is, though, that you don't need to think/know about dimensions. So, a linear map in the infin. dim case is as easy as in the other case.

What about the following rough outline:

• Some examples (they should really be first, to give motivation), including solutions of linear eq.
• Bases, dimension. Referring to the examples laid out above, by briefly "calculating" the involved dimensions. Use the first example to indicate that there is (in psychological contrast to R^n etc) in the solution space no preferred choice of generating elements, thus calling for the appropriate notion --> basis. Don't expound the infinite-dim case too much, since this is the main topic in topological vector spaces (and is, from the linear algebraic point of view, pretty much the same business).
• Linear maps. Reinterpretation of solutions as kernels of linear maps. Brief discussion of fundamental isomorphism and involved dimensions. Link this to the first example. State and emphasize that dim V characterises V up to (non-unique) isomorphism, and that maps are upon choice of bases given by matrices. Similarity of matrices. Eigenvectors etc.

Above you seem to indicate that you want to mention dimension of the tensor product. I would refrain from that, that's not so important here. Jakob.scholbach (talk) 15:14, 21 October 2008 (UTC)