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)