Talk:Kernel (mathematics)

Old discussion

I've never been very happy about the organisation of the pages on 'kernel'. Charles Matthews 09:48, 13 July 2005 (UTC)

Nor me, exactly. Though I'm not sure what the best way to reorganize would be. What do you propose? -- Fropuff 13:59, 13 July 2005 (UTC)

Compared to other math. pages, both are very difficult to understand (I think they should be merged). The algebra page should be reorganized and first present the general case (that is now at the end of the algebra page) because it is the most comprhensible formulation found in both pages. I don't know enough about maths to do this.

Keep in mind while considering this merge that there are some textbooks that use the term "null space" exclusively. As such, this would only be a good idea if there were a redirect from the "Null space" article to the "Kernel" article. 155.98.70.69 20:43, 29 March 2006 (UTC)Llyr (too lazy to log in :) )

I disagree. That is, I don't believe null space should be combined with kernel. My reason is this: the notion of kernel immediately brings to mind the concept of a homomorphism. To be precise: The kernel of a homomorphism # of a group G into a group G' is the set of all elements of G mapped onto the identity element of G' by #. For any homomorphism of a group G, the image is just G/K [read G modulo K] where K is the kernel and a normal subgroup of G. One need not go outside the group to find or produce all homomorphic images. 'K' is the kernel of the homomorphism and the identity 'element' of the factor group. K = {e'}#inverse [# to the minus 1] for #:G--->G'. As an example: the group of integers, Z, modulo some integer 'n' times Z, generates the factor group Z/nZ, a homomorphic image consisting of a group of equivalence classes isomorphic to Zn [Z sub n] [elements: 0,1,2,3,...(n-1)]. In this case, nZ is the kernel of the map. [user: adriandorn April 13, '06]

Personally, I find "null space" to be more congruent to what being defined. It is reasonably intuitive that the null space of a linear transformation A is the x's for which Ax=0. Strang uses "null space", and so do some others. I think the searcher will often look for null space. I did.

Null space and kernel should be linked, but articles should be kept separate. Gene Ward Smith 04:48, 30 May 2006 (UTC)

I have a lot of sympathies for Mergism, but in this case, it's a bit tricky. A Structurist solution might be better here. I would rather see less content and more disambiguation and sectioning on this page, with "Main article: ..." just under the heading of each section. What I don't like about this page is that it favors the set-theoretic/algebraic usage of kernel a bit too much over the other uses, such as the integral transform or statistical usages.

Although the set-theoretic usage is more general than the algebraic, it is not the case that someone trying to understand the notion in group theory or ring theory for the first time really needs the set-theoretic version. Usually, "preimage of the neutral element" will do. From this point of view, kernel_(algebra) is OK, although still a bit too chatty, particularly in the "General Case" section. (The "Mal'cev algebra" section needs help because of the redlinks.) It also underplays the category theory notion too much compared to the universal algebra notion. Again, I think that page could be improved with better sectioning.

Null space is very problematic. It is claimed on that page that some people use both that term and the term kernel to denote the zero set of an operator, where the latter term is used in its broadest, possibly nonlinear, sense. That coincides with the algebraic version only in the case of linear operators, i.e., linear transformations. I confess I have never seen either null space or kernel used synonymously with zero set, but have only seem the terms used in the algebraic (or in the case of kernel, set-theoretic) senses. Maybe I live in a bubble. If kernel really is used sometimes in this rather weak sense, then this page needs a brief mention of it. Again, the page Null space itself is rather unstructured. Better sectioning could probably fix it.

Hmm. That makes three times I mentioned better sectioning. Maybe I really am a Structurist. --Michael Kinyon 19:21, 5 August 2006 (UTC)

---

is the kernel from ordinary differential equations, the kernel used for Laplace transformation same as the kernel used for linear algebra? if not can someone clarify the first a bit more? -

I say we keep it separate, since null space is essential a primer to understanding kernel for linear algebra novices like myself.

"Kernel" defined in more ways than intro suggests?

The introduction now reads: "In mathematics, especially abstract algebra, a kernel is a general construction which measures the failure of a homomorphism or function to be injective."

Given the meaning of "kernel" in integral calculus it seems to me that either (1) this introduction would be better (not good, merely better): "In mathematics, the term "kernel" has multiple meanings. In abstract algebra, a kernel is a general construction which measures the failure of a homomorphism or function to be injective.", or (2) the relationship among the different meanings should be outlined.

Harold f 06:39, 31 December 2006 (UTC)

Convert to dab

This page should be converted into a disambiguation page. There is a common meaning of kernel in abstract algebra, linear algebra, and category theory (though not in the sense of "equalizer"). The usage of kernels in set theory, and for equalizers, seems rather rare to me, and the alternative meanings in integral operators and statistics are completely different. Geometry guy 01:26, 1 June 2007 (UTC)

Kernel cleanup (copied from Wikipedia talk:WikiProject Mathematics)

Should kernel (mathematics) become a disambig page? There was a complaint that the main usage is 4th on the list (a few pages down). If this was a disambig age, being fourth would not be a problem. Also in my experience (at a PDE heavy institution) the main usage of kernel is the one from integral operators, such as a heat kernel or green's function.

Once we have a sane disambiguation page (either Kernel (disambiguation) or Kernel (mathematics)), then we should probably go through the kernel articles and make sure appropriate links are made. I've added some links to kernel (algebra), but it is missing kernel (category theory) and probably others (hard to tell, since I think the disambig page is also missing some, and certainly kernel (mathematics) is missing many). JackSchmidt (talk) 16:28, 20 June 2008 (UTC)

I think it would be preferrable to have one good article covering all the meanings of kernel related to f(x)=0. The only article which presents substantial additional material is kernel (matrix), parts of which (such as numerical calculations) could be summarized in this article. The terrible amount of redundancy could be amended if material from sections like Kernel (algebra)#Ring homomorphisms would be moved to ring homomorphism, in this case (which, by the way, also already contains material on the kernel). Similarly, all instances of "X/ker (f) \cong im(f)" should be centered in Isomorphism theorem and so on. Jakob.scholbach (talk) 16:47, 20 June 2008 (UTC)

This seems reasonable, and would cut down on the confusion of analytic kernels versus algebraic kernels. Should the other algebraic kernel articles redirect to sections of the one true algebraic kernel, or to their associated homomorphism pages? JackSchmidt (talk) 18:07, 20 June 2008 (UTC)

Another question is what to do with the (related) category theory and set theory/universal algebra definitions. Since both are quite abstract it might be best to keep them separate, e.g. kernel (algebra), kernel (category theory), kernel (equivalence relation). --Hans Adler (talk) 18:14, 20 June 2008 (UTC)

@Jack: I would redirect to the kernel page. The homomorphism pages should just have a (reasonably brief, but complete) definition of the kernel in that specific case. Anything else (which is unspecific to groups/rings/blabla) should be put to kernel.

@Hans: Again, a good survey article should not omit the more abstract kernels. I guess, they deserve a little section in the survey article. Anything which does not fit there spacewise goes to a subpage.

@everybody: Why is the kernel in PDE also called kernel? If there is a reason (which I assume), this should be covered in the survey article, too. Alas, die Kunst ist lang und kurz ist unser Leben (art is long, and our life is short). Jakob.scholbach (talk) 18:48, 20 June 2008 (UTC)

(P.S. BTW, phrases (from the lead of kernel (algebra)) like "the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective" are probably exactly what the engineer above is complaining about. Why not write "the kernel is defined by elements mapping to zero" and treat injectivity later?) Jakob.scholbach (talk) 18:52, 20 June 2008 (UTC)

Changed into DAB, and replaced wording that would be incomprehensible to anyone but a specialist (that belongs to the individual pages, if anywhere) by something that may not be so exact but is hopefully understandable. About a year ago I have moved several nullspace articles into kernel articles, but I have stopped short of such radical cut at that time. Jmath666 (talk) 05:10, 27 June 2008 (UTC)

Entries from Kernel

Following discussion on Talk:Kernel, I've replaced the Mathematics section with a single entry pointing here. There were three entries there not on this page, but I'm not certain where they ought to go or even if they all ought to be included.

• Polygon kernel
• Kernel smoother
• Cooperative game#The kernel

CarrieVS (talk) 07:55, 7 June 2012 (UTC)