Definition talk:Vector Subspace

Looks like there is a need for the page Definition:Linear Manifold to define the concept of a closed linear subspace? --prime mover 09:50, 17 December 2011 (CST)

You may have misunderstood. A linear manifold is what is defined here as a linear subspace. Then, a linear subspace is meant to be inherently closed. I have adapted the statement for clarity. --Lord_Farin 09:55, 17 December 2011 (CST)

Having done that, I want to point out that it is less ambiguous to write closed linear subspace every time Conway (my source for this terminology) writes linear subspace. --Lord_Farin 09:58, 17 December 2011 (CST)

I wouldn't have a problem with renaming this page "linear manifold" (along with some associated rewording), then adding a new page "linear subspace" to define "closed linear manifold". The name "vector subspace" can still sit there with a redirect. My source work (Warner) is, I have been informed, unusual in its terminology and symbology, so I'm more than happy to defer to a more mainstream set of definitions etc. --prime mover 17:19, 17 December 2011 (CST)

Mainstream terminology is better, indeed. However I haven't encountered 'linear manifold' outside Conway, and hence am quite reluctant to let it prevail. Other opinions/references on this? --Lord_Farin 02:25, 19 December 2011 (CST)

When $T$ is a closed subset of $S$, isn't it automatically a $K$-vector space? --Lord_Farin 03:49, 3 February 2012 (EST)

Pm, what is your idea for Definition:Linear Manifold? Something like Definition:Linear Subspace being a transclusion to this page and referenced one?

I think that could work when this page is given an 'about' tag, something like:

'This page is about a subspace of a general vector space. Some types of vector spaces have a narrower definition of a subspace. See Definition:Linear Subspace.'

or maybe simply applying the about template:

This page is about Linear Subspace in the context of Vector Space. For other uses, see Linear Subspace.

What do you say? --Lord_Farin 17:24, 3 February 2012 (EST)

Surely it should just be as simple as "a linear manifold is a vector subspace of a Hilbert space that yadayada ... whatever the details. Also see ..." etc. but neatened up according to our HR. After all, from what I understand it's a vector subspace with extra conditions on it, same as a vector space is a module with extra conditions on it. Or is it more subtle than that? --prime mover 18:10, 3 February 2012 (EST)

It is more subtle. As I said above, a linear manifold is a linear subspace on PW; a linear subspace on a Hilbert space is a linear manifold that is closed. At least, that's what Conway says. So there is a problem with double nomenclature; hence my thoughts about a disambiguation (as nobody ever speaks about a vector subspace of a Hilbert space). --Lord_Farin 03:17, 4 February 2012 (EST)

In that case, a) A page "Linear Subspace" which has 2 sections: 1: a link to Vector Subspace, explaining that it's the same thing, and 2: A statement that it is a "linear manifold" which is closed. Then b) A page "Linear Manifold" which contains a redirect to Vector Subspace and a category indicator to Hilbert Space. My point is: if there's a term that is used, we need an entity on ProofWiki so that a user who enters it will be directed to some page that either defines it or tells the user that it means the same thing as something we have defined.

But as you point out, I'm not familiar with the details as I haven't studied any of this (I'm learning as I go, my formal education stops at an MMath). All I was originally doing was pointing out that "Linear Manifold" needed some sort of page (the nature of which I was guessing at) to achieve the above effect.

Oh, and while we are about it, we would need to make a link to it from "Manifold", either as a disambiguation or (if it's the same thing) some words explaining what their conceptual connection is. --prime mover 03:56, 4 February 2012 (EST)

And, apologies, we already discussed this in December, I completely forgot (early onset alzheimers). I hope I've been consistent at the very least. --prime mover 04:13, 4 February 2012 (EST)

I think it's done. Many pages need to be adapted to link directly to the appropriate material, though. --Lord_Farin 05:58, 4 February 2012 (EST)

Linking done. --Lord_Farin 06:14, 4 February 2012 (EST)

Works for me. Thx for insights in the work on Zorn's Lemma by the way. --prime mover 07:05, 4 February 2012 (EST)

Non-Empty?

Which of the criteria for the definition here makes sure that the subspace is not empty? I learned that by definition a linear subspace has to have an element, is that implied here? --GFauxPas 14:53, 11 March 2012 (EDT)

Yes. It is stated that a subspace is a vspace itself. Then, a vspace is a group under addition, hence nonempty. --Lord_Farin 19:00, 11 March 2012 (EDT)

Re: whether the definition I added for $\R^n$ is a result or whether it's a definition: Khan and Fraleigh both have it as a definition. I don't have enough understanding of the general definition at the top of the page to determine how the $\R^n$ definition plays into the grand scheme of things. If One-Step Vector Subspace Test is both necessary and sufficient, then it looks like the definitions would be equivalent, but that page only has it as a sufficient condition. --GFauxPas 19:01, 12 March 2012 (EDT)

We've had a similar conversation before. If there are several different ways of defining an entity, and all are equivalent, then we can do one of two things:

a) Write a large page (complete with transclusions, if need be) and cite all the definitions, then prove piecemeal that they are all equivalent

b) Pick one definition (the most intuitively straightforward, IMO), prove that all the other statements are equivalent to it, and then mention that "some sources use this as the definition".

b) is the way a lot of the pages have been structured. It makes it easier to manage.

Incidentally, as your new definition only covers the specific instance of $\R^n$ and not the general vector space, I'd be prepared to discount it as a definition. Fraleigh and Khan only treat the conventional real-number space (it's all that's needed in elementary applied maths and physics, and that's where those texts are directed) so that's what they define.

So, IMO, this statement is safest as a result not a defn. --prime mover 19:09, 12 March 2012 (EDT)

Agreed, but then it's years ago I learned about 'the grand scheme'... Nonetheless, firmly agreed. --Lord_Farin 19:12, 12 March 2012 (EDT)

I'm having a circularity problem. $\mathbb{W}$ is a subspace, so it's non empty, and that's how I know it contains the zero vector, because it's closed under multiplication by zero scalar. $\mathbb{W}$ is a subspace because it passes the One-Step Vector Subspace Test. One of the conditions for the Two-Step Vector Subspace Test is that $\mathbb{W}$ is non-empty. I know it's non empty, because it contains the zero vector. he;p --GFauxPas 19:31, 12 March 2012 (EDT)

It's circular because of the inference '$W$ is a subspace as it passes One-Step Vector Subspace Test'. You chose an involved way of saying: 'it passes the subspace test as it passes the subspace test'. This is bound to be interpreted as circularity, because it is. --Lord_Farin 19:38, 12 March 2012 (EDT)

Fair enough. I gave the $\R^n$ stuff its own page. --GFauxPas 19:44, 12 March 2012 (EDT)

Good stuff. I renamed it to match the definition we already have for this item (refactoring in progress). --prime mover 03:10, 13 March 2012 (EDT)