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February 23

Left null space and cokernel

According to the article on the four fundamental subspaces, the left null space is the cokernel, but according to the section on left null space, it is dual to the cokernel. Which one is correct, and also, how is the definition $\mathbf {x} ^{T}A^{T}=\mathbf {0} ^{T}$ related to the quotient space definition of the cokernel?--Jasper Deng (talk) 17:30, 23 February 2017 (UTC)[reply]

See p. 27 here [1]. I think some wires must have gotten crossed, due to the various notions of "is", and how careful (or not) authors try to be when maintaining distinctions between isomorphic things with similar symbols and names. The way many authors define it, the left null space is a simple subset of the codomain, whereas the cokernel is a quotient space. So, one can say that the left null space is isomorphic to the cokernel, but it's not terribly wrong to say it simply is the cokernel, in anything but the most formal contexts (all we have to do is make a natural identification between elements of the codomain and elements of the quotient space). SemanticMantis (talk) 18:40, 23 February 2017 (UTC)[reply]

@SemanticMantis: But certainly being dual to is more specific than just "isomorphic to". In the finite-dimensional case, being isomorphic simply means being of the same dimension over the same field; I'm guessing what's being looked for here is a "natural" isomorphism of sorts.--Jasper Deng (talk) 19:29, 23 February 2017 (UTC)[reply]

Yeah, I think that dual statement might just be wrong, but I'm not sure. Do you have access to a university library? This is a case where a real textbook can clear things up far faster than combing through lots of online refs. If it's not wrong, then I think the dual statement must be assuming some different definitions than those used in the other articles. I'm not even sure what kind of "dual" it would be. But I'm very rusty on this stuff, hopefully someone else will chime in. SemanticMantis (talk) 22:54, 23 February 2017 (UTC)[reply]

I actually do have access to such a library (in fact I'm in one as I speak). I have in my possession an upper-division linear algebra book, but I don't think that one covers the left null space and cokernel, so I probably will want to dig in the library.--Jasper Deng (talk) 22:57, 23 February 2017 (UTC)[reply]

User:Jasper Deng did you find some clarifying definitions? If you do, let me know, and I/we can clarify our articles. SemanticMantis (talk) 18:00, 25 February 2017 (UTC)[reply]

I haven't had a chance to look in the library just yet. It may be a little while before I can.--Jasper Deng (talk) 19:55, 25 February 2017 (UTC)[reply]