Talk:Rank–nullity theorem

Rank theorem

Why does "Rank theorem" redirect to this article?128.135.239.238 (talk) 02:20, 23 April 2008 (UTC)

I presume because the phrase "Rank theorem" is rather similar to "Rank-nullity theorem". In addition, a Google search of "rank theorem" reveals that this phrase is used to refer to the rank-nullity theorem in at least some informal capacity. But you seem to be implying you have a better page for it to redirect to?173.195.7.148 (talk) 15:02, 7 December 2012 (UTC)

Proof

Can't u show all the process of proving the relation : rank(T)+Nullity(T)=Dim(V)

-- I've posted a proof of this (DriveOnTheAutobahn/128.253.69.185)

The proof (notation mostly), should be modified so that it works for infinite dimensional vector spaces, at least, this should be mentioned, since it is in the introduction. Lewallen (talk) 00:20, 28 February 2009 (UTC)

Infinite-dimensional

Surely V can't be infinite dimensional? How then would it's rank be well defined? —Preceding unsigned comment added by 152.78.171.187 (talk) 20:58, 1 June 2009 (UTC)

Of course, $V$ can be infinite dimensional. $rank(T)$ is then still well defined in the same old way:$rank(T)=\dim(im(T)))$ Still $im(T)$ is a subspace of $W$ and as such it is a vector space and has a dimension $\dim(im(T))$, which is the cardinality of any of its bases. The theorem still holds true since the addition of cardinalities of disjoint sets is just the cardinality of the union.

Who proved this?

Anyone know any of the history of this result?? —Preceding unsigned comment added by 143.234.196.119 (talk) 16:15, 4 December 2009 (UTC)

never seen rk, does it stand for rank? — Preceding unsigned comment added by 79.115.169.113 (talk) 05:09, 16 July 2014 (UTC)

Nomenclature

Nomenclature looks wrong/ambiguous (not defined in article).
Specifically, shouldn't  nul(A)  be  dim(nul(A)) ?
—DIV — Preceding unsigned comment added by 120.17.136.99 (talk) 07:38, 27 August 2016 (UTC)

nul(A) is fine; the nullity is defined as the dimension of the kernel: nul A = dim(ker A). Joel Brennan (talk) 16:54, 22 April 2022 (UTC)

Terminology

This article relies too much on non-elementary symbolic operators. The symbolic statements should be accompanied by English versions of the ideas expressed, so that novice readers can decipher them by following links from the English words. The article states the theorem in language that may be unfamiliar to anyone who is unfamiliar with theorem -- in other words, the only people who can easily understand this article are people who probably already know what it's going to say.

At first glance I see: Hom(...), dim, Im, Ker., ≅ (as used here), := (as used here), $\mathbb {F}$, etc. ---- 69.193.134.179 (talk) 20:46, 13 August 2018 (UTC)

I agree, and I have tagged the article accordingly. D.Lazard (talk) 17:13, 22 April 2022 (UTC)

Proposed merge of Fundamental theorem of linear algebra into Rank–nullity theorem

PRODded as a duplicate article under a nonstandard term, but a user objected and suggested merger. This makes sense, since this article is more clearly written then the other one, which is too heavy on symbolic math. –LaundryPizza03 (dc̄) 13:07, 17 April 2023 (UTC)

• Support per nom. D.Lazard (talk) 14:05, 17 April 2023 (UTC)

  I have done the merge and converted FTLA into a redirect. I did not include the transposed version of the statement at Fundamental theorem of linear algebra in this article, and anyone who cares is invited to check and see if there's something work bringing. I also omitted the citation Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388, anyone with access to that source is invited to add it (but it would be nice to add the appropriate location in the book to the citation). --JBL (talk) 00:05, 20 April 2023 (UTC)

Extension to infinite dimensional spaces

There is a very central theorem in functional analysis where if we have a bounded linear operator between Hilbert spaces T with adjoint T*, we have that the orthogonal complement of the range of T is the kernel of T*. Similarly, the orthogonal complement of the kernel of T* is the closure of the span of T. I am unaware if this theorem has its own page. If not, it sufficiently central that it ought to be somewhere (the finite dimensional version is very briefly stated without backing here https://en.wikipedia.org/wiki/Kernel_(linear_algebra) in terms of quotient spaces). This page might be a good home as the rank-nullity theorem is its immediate consequence in finite dimensions. It is also mentioned in terms of a "cokernel" in the section "A third fundamental subspace" which is a great start but I find this paragraph not the most revealing to say the least. It could go in the generalizations section, but do people agree? Scienceturtle1 (talk) 05:52, 5 July 2024 (UTC)

Although I do agree that this theorem should be mentioned somewhere, I doubt this is the proper page to do so, as this mainly discusses the finite dimensional cases. Certainly, I would think that this theorem deserves its own proper discussion, and I do not think that merely mentioning it here will do it justice. I am quite inexperienced as an editor, so feel free to correct me, but I think it best gets its own space. BrakkoFTW (talk) 09:41, 14 August 2024 (UTC)

Diagram illustrating subspace decompositions

Dear JayBeeEll, You are correct on all counts.

We would still like to contribute a visual that fits better with the article, if it can be adapted appropriately. We had a couple of questions:

1. Do you think a diagram like this would be more appropriate on the "Row and column spaces" page, where all four subspaces are discussed more directly?

2. We can revise the figure to better match the notation used in the article. Would reducing or removing most of the explanatory text (keeping only labels) make it more suitable for Wikipedia style?

3. We would appreciate any guidance on how to adapt it so that it is clearer and better aligned with the article.

Best regards, Eteri Byazrova & Yuri Morozov Eteri byazrova (talk) 20:03, 19 March 2026 (UTC)

Hi @Eteri and Yuri,

In answer to the first question, yes, it does seem like a diagram along these lines would be better suited to that article. For the other two questions, illustrating mathematics is not one of my personal strengths, so let me point you at some resources to consult rather than give you my individual thoughts:

• MOS:IMAGES is the main Wikipedia Manual of Style page dealing with images
• Wikipedia:Manual of Style/Mathematics § Graphs and diagrams
• Wikipedia:Manual of Style/Accessibility § Images
• WT:WPM is a discussion page for any issues having to do with mathematics on Wikipedia; you could seek more opinions or discussion there.

I hope that this is somewhat helpful, and thanks for your contributions to Wikipedia! --JBL (talk) 22:54, 22 March 2026 (UTC)

Hi @JBL,

Thank you — this is very helpful, and we appreciate the pointers to the style guidelines.

We’ve adjusted the image placement accordingly.

If you ever need help with creating or refining diagrams, that’s something we work on extensively and would be happy to contribute.

Thanks again for your guidance and for your work on Wikipedia.

Best regards,

Eteri Byazrova & Yuri Morozov Eteri byazrova (talk) 23:35, 22 March 2026 (UTC)