Talk:Kernel (linear algebra)

Linear operator -> linear map

This article is about the kernel of a linear map. A linear map is called "linear operator" only in functional analysis, which is not the subject of this article. The problem lies in the fact that for most people and most WP articles using the concept of kernel, "linear map" is the standard name. I have encountered this problem by trying to disambiguate the links to kernel (mathematics) and finding that kernel (linear algebra) does not exists.

Thus I have edited the article to replace "operator" by "map" and I will move it to kernel (linear algebra) D.Lazard (talk) 11:47, 28 September 2012 (UTC)

Merger

I propose to merge this article with Kernel (matrix). I think the reasons are quite obvious to anyone familiar with the subject. The two concepts are so intimately related it is often beneficial to conflate them (in the finite-dimensional case, that is) or treat a kernel of a matrix like a special case of a kernel of a linear map. I also expect sources to discuss one in conjunction with the other. Keφr (talk) 19:59, 26 November 2012 (UTC)

• Support: The kernel of a matrix is the kernel of the linear map it defines. In kernel (matrix), it is implicitly defined as such, even this is made explicit only later in the article. It seems that the editors of this article wanted to be more elementary by not talking of linear maps and vector spaces. IMO, this is silly, as the kernel is a vector space, and forgetting this remove any interest to the notion. Similarly, something that looks as a matrix without representing a linear map is not a matrix, but an two dimensional array. D.Lazard (talk) 08:08, 27 November 2012 (UTC)
• Support: for merging under this article's title, Kernel (linear algebra), rather than Kernel (matrix), since the former is the general concept, and the latter is merely a representation. — Quondum 09:43, 27 November 2012 (UTC)
• Support: there are only one concept (aside from integral kernel); therefore, there should be one article on it. -- Taku (talk) 11:15, 3 December 2012 (UTC)
• Performed by Keφr 11:19, 29 September 2013 (UTC) (Almost a year. Wow.)

Improvements

I respond to the request to clean up the problem "Decide whether to explain this in terms of linear maps or matrices, or preferably organize the different views. Remove redundancies. Clean up markup. Do something with short sections."

• I cast the matrix A as a representation of the mapping. This should make clear that multiplication by a matrix over a vector field z is a representation of a linear map, not exhaustive of the class of linear maps.
• I changed "null space" to "kernel". If the article should instead use the term "null space", then perhaps the article title needs to be changed accordingly.
• I gathered all definitions of terms that were independent of a matrix representation and separated out those that were being discussed entirely in a matrix multiplication context.
• I refined the section now called “Illustration”.

ArthurOgawa (talk) 04:18, 19 October 2014 (UTC)

I think the {{cleanup}} template may now be removed. There still remain several {{citation needed}} templates to be dealt with. ArthurOgawa (talk) 21:14, 19 October 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Kernel (linear algebra)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs references and more information on the connections with functional analysis. Jim 18:49, 9 September 2007 (UTC)

Last edited at 18:49, 9 September 2007 (UTC). Substituted at 02:16, 5 May 2016 (UTC)

Nomenclature

The term "nullspace" seems to have been unfairly depricated here. "kernel" is widely and ambiguously used outside of the math subfields, with different connotations particularly in modelling and computation. Although there is a natural reason for this overlap because of Green functions for the heat equation correspond to both nullspaces and statistical distributions useful in smoothing, partial differential equations are not necessary for the understanding of either on their own. In the current context, the term "nullspace" suffers no such difficulties, and benefits for iconicity relating the concept immediately back to linear algebra theory where most students are first exposed to the concept. — Preceding unsigned comment added by Uscitizenjason (talk • contribs) 16:34, 19 March 2017 (UTC)

"Nullspace" is not deprecated here. The article's title comes that, in modern mathematics, kernel is presently more widely used, probably (this is a personal guess), because of the dual concept of cokernel (the duality between null space and quotient space is not clear from the terminology). However, a hatnote is lacking, which I'll add. D.Lazard (talk) 17:52, 19 March 2017 (UTC)

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Columns in "Computation by Gaussian elimination" section

Why is this section stated and demonstrated using column-based Gaussian Elimination? That is, normal Gaussian elimination allows you to make sums/multiples of the rows of the matrix. Yet the example given here uses sums/multiples of the columns instead. This could be highly confusing to a reader, especially when they visit the link to Gaussian Elimination all theory and examples are row-based.

Not to mention that the formatting of the page would be much nicer with matrices looking like $\left[A|I\right]$ and $[B|C]$ instead of the vertical matrices $\left[{\frac {A}{I}}\right]$ and $\left[{\frac {B}{C}}\right]$.

Also, given that this method is similar to finding a matrix inverse, it would be nice if this was consistent with that too.

"NS(A)" listed at Redirects for discussion

The redirect NS(A) 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 July 21 § NS(A) until a consensus is reached. Steel1943 (talk) 21:16, 21 July 2023 (UTC)