Talk:Fundamental theorem of linear algebra

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Rewrote the page in terms of the SVD. — Preceding unsigned comment added by 171.67.87.126 (talk) 21:07, 21 June 2011 (UTC)

What is the notation "U^+" used in the article (under the basis column)? Thanks. 99.236.122.76 (talk) 03:47, 22 May 2011 (UTC)

Well, this is certainly an old-fashioned way of discussing something - not exactly clear in the notation: probably the general linear mapping, and its effect on the dual spaces. It will need some reconciliation with the rest of the linear algebra pages.

Charles Matthews 07:01, 8 Oct 2004 (UTC)

Oh, just for the record, I wrote that table myself and did not just copy it out of a textbook, so no copyvio issues here. —Lowellian (talk)[[]] 18:48, Oct 9, 2004 (UTC)

I think this needs researched just a bit. The August 2005 issue of Focus by the Mathematical Association of America discusses some candidates for the Fundamental Theorem of Linear Algebra. Whatever textbook this theorem came out of is quite possibly an author's whim and not a generally accepted term.

It's Gilbert Strang who calls it that, but he seems to have convinced other people to call it this as well. Swap (talk) 20:13, 20 August 2009 (UTC)

Not everyone. I think it's silly. Zaslav (talk) 03:41, 15 April 2023 (UTC)

Column space/LDU decomposition

Shouldn't the column space be related to the columns L. Something like: P^{-1} times the first r columns of L?

Further: is it sensible to use LDU to define the four subspaces? Not better the SVD? One problem in the article is, that it states, that entries in D are non-decreasing, but that doesn't make sense, since the entries can be positive and negative. What is wanted here is certainly that the nonzero ones come first and then all the zeros. The SVD would clearly make this easier, obviating also the need for a permutation matrix. 134.169.77.186 (talk) 10:09, 9 May 2011 (UTC)

Opaque

This article seems very opaque, even when I already know something about the subject - perhaps something could be done to make it more friendly to those who do not know a lot about the subject already? — Preceding unsigned comment added by 142.151.247.134 (talk) 04:55, 29 February 2016 (UTC)

I agree. The fundamental theorem is not even stated. Moreover, the article involves orthogonal complement, which implies that the fundamental theorem does not exists for vector spaces that do not have a dot product. I guess that the fundamental theorem is nothing else than the isomorphism theorem for vector spaces. I have tagged the article for warning the readers of this mess. D.Lazard (talk) 11:31, 29 February 2016 (UTC)

Three years later, I have stated what seems the intended theorem. The remainder of the article remains a mess. It is full of inaccuracies, and requires to be expert of the subject for understanding the intended meaning. Therefore, I'll probably remove most of the content.

Move, merge or delete?

Recently,this article was a mess [1], as, it was almost impossible to extract from it the statement of the theorem referred to in the title. I have edited it for reducing it to the statement that can be guessed from this previous form.

By doing this, I have removed the part of the article that refers to an inner product, because, otherwise, this would suggest that linear algebra without inner product is not fundamental.

Although the results that remain in the present state of the article are important, and deserve to be accessible in WP, this seems excessive to qualify them as the fundamental theorem of linear algebra.

Firstly, these results are corollaries of the existence of bases for every vector space, and therefore cannot be qualified as fundamental.

Secondly Scholar Google gives 715 hits for "fundamental theorem of linear algebra" while there are more than 1.7 million of hits for "fundamental theorem of algebra", and 198,000 hits for "fundamental theorem of projective geometry". Thus, "fundamental theorem of linear algebra" is not a common name.

So, this article should be either moved, merged or deleted. I have not found any plausible target for a move or a merge. Therefore, for the moment, I would be inclined to nominate the article for deletion. Before doing that, I'll notify this thread to WT:WPM, in view of better suggestions. D.Lazard (talk) 15:10, 22 March 2020 (UTC)

We could merge it into Gilbert Strang; I gather that it's part of the way he teaches linear algebra, and introductory texts on linear algebra are a big part of what he's known for. XOR'easter (talk) 15:46, 22 March 2020 (UTC)

I would have expected something with this name to be the theorem that every vector space has a basis, but we don't have a separate article on that; instead it's buried deeply within Basis (linear algebra). Should it be a standalone article? And if so, what should it be called? In any case, what's left here is something only about finite-dimensional vector spaces, important but not fundamental to all of linear algebra. —David Eppstein (talk) 17:33, 22 March 2020 (UTC)

I agree with the points above. In particular, the term "fundamental theorem" does not seem widely established, and this theorem is barely more than rank-nullity, which is an established name.

Further, many linear algebra books have catch-all theorems summarizing how basis, span, linear independence, rows, columns, determinant, etc. all relate to each other, and the "fundamental theorem" as stated here leaves out several of those sub-theorems. Finally, it's arguably POV to state a result about matrices as fundamental to linear algebra, rather than a result about vector spaces or linear transformations. And it's arguably POV to bring up inner products in the guise of transposes.

So I vote that this article can be merged into Rank-nullity theorem, Basis (linear algebra), etc. Mgnbar (talk) 18:44, 22 March 2020 (UTC)

As there is a clear consensus that "Fundamental theorem of linear algebra" is not a widely established name, I have removed all incoming links, and retargeted Four fundamental subspaces to Kernel (linear algebra)#Left null space. So the article is now an orphan. D.Lazard (talk) 10:43, 23 March 2020 (UTC)

My impression is that the article suffers from the notability requirement; the notability, not of the mathematical facts but of the name. So, after merging content to the other aritlces, as needed, I think the article should be deleted; perhaps deletion by a redirect. -- Taku (talk) 21:37, 23 March 2020 (UTC)

So that beautiful image is gone.-.- file:The_four_subspaces.svg — Preceding unsigned comment added by 80.99.107.132 (talk) 11:56, 26 March 2020 (UTC)

"Beautiful" is a valuable point of view that I do not share. In any case, this image is highly confusing, as the lozenges are not explained, and the inclusions between spaces are not indicated. D.Lazard (talk) 12:52, 26 March 2020 (UTC)

I understand that it wasn't explained clearly, but is it mathematically correct? If someone were to provide further explanation, could it be readded? — Preceding unsigned comment added by 80.99.107.132 (talk) 18:43, 3 April 2020 (UTC)

Proposed deletion

Merge and delete: it is silly to have an article for Strang's personal foible. I agree with Mgnbar and David Eppstein. Zaslav (talk) 03:45, 15 April 2023 (UTC)

Looking at Rank-nullity theorem, it seems this article has nothing essential. The cokernel property should be added to Rank-nullity theorem if worth stating. I am proposing deletion. Zaslav (talk) 04:13, 15 April 2023 (UTC)

I don't think deletion makes any sense, but a merge and redirect seems sensible. --JBL (talk) 21:02, 19 April 2023 (UTC)

Ok, I have done a partial merge, I will turn this into a redirect. --JBL (talk) 23:59, 19 April 2023 (UTC)