Talk:Direct sum of modules

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Mathematics rating:
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Field: Algebra

Other direct sums

Although the text refers to the direct sum of abelian groups, it does not discuss the direct sum of non-abelian groups (let alone semi-direct sums or subdirect products, etc. of same).

Although as time has gone by, I've learned enough to enjoy the "big picture" of algebra, I needed something a bit more specific when I was going around the block the first time. And category theory is still a bit far off in the mists for me.

Perhaps in this case, a "direct sum of groups" page should be added which can back link to this page, which could be retitled something like "Algebraic direct sums" or just stay "direct sums". I could make a similar comment about direct product which is currently (mostly) a "direct product of groups" page. Any thoughts? chas 19:39 7 oct 02 (UTC)

As I mentioned in one of the edits, "direct sum of groups" really should redirect to "free product of groups". Revolver 18:32, 30 August 2005 (UTC)

Linear algebra

I've only taken a linear algebra course so I found the article rather incomprehensible. My definition of a direct sum goes like this: Let V be a v.sp. and U1,...,Um be subspaces of V. Then V is a direct sum denoted V = U1 (+) ... (+) Um iff each v ∈ V can be written uniquely as a sum u1 + ... + um st each ui ∈ Ui.

Comments welcomed! Goodralph 23:20, 11 May 2004 (UTC)

That's an internal direct sum; you are already provided with the ambient V. Charles Matthews 05:45, 12 May 2004 (UTC)

Direct sum & Cartesian product

I'm a bit confused as to the relationship between the direct sum and the Cartesian product. It appears that if I have two abelian groups, $(G, *_G)$ and $(H, *_H)$, I can take the direct sum of those groups to be

$(G, *_G) \oplus (H, *_H) = (G\times H, *_{G\oplus H})$

where $*_{G\oplus H}$ is defined by

$u *_{G\oplus H} v := (u_G *_G v_G) \oplus (u_H *_H v_H)$.

Is that correct? If so, should "direct sum" be mentioned on the Cartesian product page? —Ben FrantzDale 14:24, 2 January 2007 (UTC)

seems that the issue is just semantics. as your notation indicates, the underlying set is the Cartesian product of sets, on which you define an operation, and call this pair the direct sum of the two modules (or the direct product of groups, etc). On the other hand, one also speaking of the Cartesian product of two algebras, where the algebra operations are defined in a similar way on the Cartesian product of sets. In that context, the direct sum refers to just the resulting module structure. Mct mht 23:23, 4 January 2007 (UTC)

Quality

This article is still rated as Start Class above and I agree that it leaves much to be desired. But there is little active editing - just some edit warring over tags now. I am restoring the tag relating to the need for footnotes which the article lacks completely. Other editors are invited to discuss their plans for improvement of this article. Colonel Warden (talk) 12:18, 25 July 2008 (UTC)

The tag is fine for now. I'm surprised there has been any "warring" over it. There hasn't been much active editing of this article for a long time. The basic core of the article is unchanged from 2002. --C S (talk) 14:04, 25 July 2008 (UTC)
Yeah, I did not remove it, since eventually it will be addressed, possibly when the article is pushed to B-class, but more likely after it has sat as B-class for a while. This is a pretty basic topic and the article gives pretty minimal and very standard coverage, so any "inline citation" in the current article is found by opening one of the references and looking for direct sum in the index, or even in the table of contents. The tag is just superfluous right now. JackSchmidt (talk) 14:12, 25 July 2008 (UTC)
Jack, consider the possibility that you're wrong about the need for inline citations. Perhaps you are not exempt from the typical arrogance of mathematics editors on Wikipedia. After all, even though many articles such as theatre lack proper (or any) sourcing, these are real topics that have been written nicely by well-meaning editors. Math articles on the other hand are typically on obscure topics of no real interest to many people and the math writers like to glorify their Ph.D. knowledge to other such experts. --C S (talk) 14:21, 25 July 2008 (UTC)
What do you mean? The article needs inline citations for GA status. It is a crap article right now. The article is a start, but it needs at the very least:
• A history section
• An examples / motivation section
• A generalisation section (direct summand and pure submodule)
• Images would be worth another 50k of text
• The lead has to be rewritten to include a summary of the article
• The sections currently present need to be rewritten into summary style
• The bulleted lists need to be made into prose
Yes it needs inline citations, but it needs so much more than that. To be clear, this is nowhere near our worst article. We have about ten thousand worse than this, all needing more work than this. Our articles need a ton of work. The point of tags like {{inline}} is to direct the editors to the most efficient use of their time on the article. I check the {{failed verification}} and {{disputed}} tags daily at WP:WPM/CA and help with that. Usually the tags are correct, but sometimes people place those tags when the article just needs copyeditting. My point is: this article needs a lot of work, because it is start class, and emphasizing the need for inline citations is premature. JackSchmidt (talk) 14:30, 25 July 2008 (UTC)
• I usually find the process of searching for specific sources helps improve the general content because you turn up unexpected details. Writing to the sources also helps reduce OR and synthesis though these may perhaps be less of an issue with maths topics. Colonel Warden (talk) 20:39, 25 July 2008 (UTC)
Certainly it would have been more efficient had the original article included the inline citations as it was written. When you have some of the references in hand, you can sort of see the verbatim lifting going on, but they do come in chunks. In some sense, this article is the opposite of WP:SYNTH! However, I agree, inline citations would make that more clear. Imagine: (MacLane 1991, p 241) blah blah (MacLane 1991, p 241) blah blah (MacLane 1991, p 242) blah blah (MacLane 1991, p 244) Image:Figure from MacLane 1991 p 243.
I think now though, the more efficient way to improve the article is to address the big concern: What is this stuff for? (motivations, history, applications, examples all basically answer this question). I think you mentioned such a concern somewhere before. JackSchmidt (talk) 21:12, 25 July 2008 (UTC)

Vector space terminology

I changed "The cartesian product V × W can be given the structure of a vector space K" to "The cartesian product V × W can be given the structure of a K-vector space", thinking it uncontroversial, as I have never seen the former wording used before, and found it actually confusing (the K is in apposition to the "vector space"). That was reverted so I've changed again to "vector space over K", which is pretty standard I think. Richard Pinch (talk) 06:26, 11 October 2008 (UTC)

"Direct sum" of algebras

The procedure described for constructing the "direct sum" of algebras actually gives the product in the category of algebras. Thus it should really be called direct product of algebras instead. When one hear the "direct sum", one expects the coproduct in the category, which is the free product in the case of algebras, and not the direct product. The terminology "direct sum of algebras" should therefore be avoided, although it is very widespread, especially in Lie theory. I suggest these things should be pointed out, including perhaps a proof (for people in doubt) that the direct product of two algebras does not satisfy the requirement of a coproduct in the category of algebras. Fistel (talk) 18:42, 22 April 2010 (UTC)

Thank you Fistel for the remarks on category language. Your frank statement that the terminology is "very widespread, especially in Lie theory" admits that this encyclopedia reflects common usage. Your note concerning "one expects the coproduct in the category" shows your concern for a particular kind of reader, not the general reader. Your note is important as mathematics communication evolves on the basis of attending to such matters. For the moment I am putting Joseph Wedderburn as a reference to confirm the present usage.Rgdboer (talk) 20:42, 22 April 2010 (UTC)

Direct sum of a subspace and its orthogonal complement?

I was reading an article by Nashed&Wahba(1974) "Generalized inverses in reproducing kernel spaces..." in SIAM J.Math.Anal., #5 — where the authors consider a linear operator A between Hilbert spaces X and Y. They immediately make the following claim: "It is well known and can be easily shown that if A is a bounded operator, or if A is a densely defined closed operator, then [generalized inverse] A exists on ℜ(A) ⊕ ℜ(A), where ℜ(A) is the range of A." So I don't seem to understand the difference between this direct sum ℜ(A) ⊕ ℜ(A), and the entire space Y — could someone clarify this?  // stpasha »  04:39, 25 June 2010 (UTC)