Talk:Basis (linear algebra)

Some points

• "every vector in V can be written uniquely as a finite linear combination of vectors in the basis". Is it necessary to say finite here?
• "Every nonzero vector space has a basis". Why non-zero? The zero space has the empty set as basis.
• "Every nonzero vector space has a basis". Strictly speaking this requires Axiom of Choice.
• "and in fact, infinitely many different bases". The first clue that these are real vector spaces.
• "every finite dimensional vector space can be considered to be essentially 'the same as' a Euclidean space". I disagree, Euclidean space has a metric, which general vector spaces don't, and the different coordinate mappings will induce different metrics back on the vector space.
• It needs to be stated that the number of elements in a basis is always the same, hence the definition of dimension.

Richard Pinch 19:43, 25 November 2008 (UTC)

Heh, I was still revising the page, so you beat me to the punch.

• I used the word "finite" not because it is necessary (which it isn't), but to emphasize the fact that bases can be infinite sets but that infinite sums are not a permitted construction in general vector spaces. I think it is useful here, but some may disagree.

• This is assuming you define the span of a set to be the intersection of all subspaces containing the set, and a basis as a linearly independent spanning set. Or, if you use my definition of basis, it assumes you define the "empty linear combination" to be the zero vector . Aiming this article at someone with little or no exposure to vector spaces, I tried to avoid this pathological case. As written, non-zero is still correct, and perhaps somewhere further down the article, it can be mentioned that it is useful to consider the empty set as a basis for the trivial space.

• True about the axiom of choice, and I did forget it -- but again, not something to mention to the non-initiate. It should be mentioned on the "advanced" subpage.

• Yes, I am implicitly assuming these are real (or complex vector spaces). In my mind, the average university-educated non-specialist is not going to look up basis trying to learn about a one-dimensional space over Z_2. On the other hand, something that must be emphasized to students just learning linear algebra (and hence, to any who aren't experts) is that generally, bases are not canonical (although certainly standard bases for various spaces should be mentioned further down the article).

• Sure, Euclidean space has additional structure beyond its vector space structure. But this is a page about vector spaces. Recognizing this difference, I put "the same as" in quotes. You could also say that the complex numbers are not the same as R^2, because you can multiply complex numbers naturally, but the corresponding construct on R^2 looks contrived. However, I have heard experts say colloquially "thinking of C as R^2...". Perhaps there could be a section mentioning inner products, and in that section, mention that R^n has a distinction from general vector spaces arising from the usual metric on R^n. I put this in because I think many texts do not emphasize enough how much similarity a vector space of dimension n has with R^n. I view correspondence of the algebra of endomorphisms of a finite dimensional vector spaces with a matrix algebra as the climactic point of an introductory linear algebra class (excepting some cool stuff once you introduce inner products).

• Yes, the independent dimension property was going to get mentioned when I revised further (of course, without using that cumbersome terminology).

It seems to me that we may have a difference in philosophy as to how articles should be written. I like to follow Halmos's advice (certainly a great expositor): "There is a difference between misleading statements and false ones; striving for 'the clear reception of the message' you are sometimes allowed to lie a little, but you must never mislead." (from I want to be a Mathematician, an Automathography, p.113). I would place all of (1)avoiding the trivial vector space, (2)avoiding mentioning the axiom of choice, (3)implicitly assuming scalars are an infinite field, and (4)saying that finite dimensional vector spaces are "the same as" euclidean space in the "lying, but not misleading" category. So first, do you agree with Halmos and I, and second, would you classify any of these topics as "misleading" instead? In any case, thanks for your vigilant and pertinent comments.Barry R. Smith 20:41, 25 November 2008 (UTC)

Well, vector spaces over GF(2) are pretty important to people in communication theory, but my point was that if you want to restrict to describing real vector spaces, that needs to be stated somewhere. De gustibus non est disputandum, but I do think that it is misleading to describe real vector spaces in terms of Euclidean spaces: I'ld prefer to equate them to Rⁿ, instantiated as, say, row vectors or n-tuples of real numbers, decribed as "coordinates" if you will. Richard Pinch 22:11, 25 November 2008 (UTC)

I would guess that for a supermajority of people who use vector spaces over GF(2), knowledge of said spaces came after knowledge of real vector spaces, but I could be wrong. Whether the assumption that scalars are real is made seems more a matter for the vector space page, where the assumption is implicit in the current introduction to the page. Certainly, mention should be made on that page that the scalars are probably most often real numbers, but don't have to be. I am not sure where on a page about bases that this should also be mentioned.

If we write it very carefully, we can avoid making implicit assumptions about the scalars (so for instance, replace the declaration that there are infinitely many choices of basis with the statement that there are usually multiple distinct bases). I worry that the reader with no knowledge of finite fields seeing this would say, "hmmm, why does it say 'usually'?", and get confused. Perhaps a good compromise on this particular issue would be to lie a wee bit, and say that there are always multiple choices of basis for a nonzero vector space.

I'm fine with your critique of the choice "Euclidean space". I'll change it when I find some time. How about the other 3 issues -- "misleading" or just "little lies"?Barry R. Smith 22:25, 25 November 2008 (UTC)

OK, that was the main point. I'm not too fussed about the others. Richard Pinch 22:30, 25 November 2008 (UTC)

Disambiguation

I suppose this page should be moved to "basis (vector space)" so as to distinguish it from, say, "basis (topology)", etc. I would guess the linear algebra usage is the most prevalent, but "basis (vector space)" seems more descriptive and just about as descriptive as "basis (mathematics)" to me. —The preceding unsigned comment was added by Barry R. Smith (talk • contribs) 06:09, 27 November 2008 (UTC)

Yes, I think so too. Richard Pinch 07:17, 27 November 2008 (UTC)

Basis (topology) exists. Richard Pinch 07:33, 27 November 2008 (UTC)

I was just about to move the page, but first some things should be clarified. My plan was to have the main page just cover bases of vector spaces encountered in introductory courses, either just over the reals, or maybe the complexes or Z_2. I would then have a discussion about bases of vector spaces over division rings on the advanced page. At that point, a discussion of bases of free modules might as well be done.

If I move the current page to "basis (vector space)", then that isn't very accurate for describing the bases of free modules in the advanced subpage. Perhaps it is better to only describe the more general concept of vector space in the advanced subpage of "vector space". I would then leave the advanced subpage of "basis (vector space)" empty, and create a page about free modules only that would include the definition of basis, and the fact that a module over a division ring is always free. We could finally have a link from the advanced page "vector space" to the page "free modules" saying that the latter concept is a generalization of the former. Sound good?Barry R. Smith 19:32, 27 November 2008 (UTC)

All these different sorts of bases are in some sense the same. More than just disambig. Peter Jackson 18:44, 27 November 2008 (UTC)

Which is why I wanted them to all be in the same cluster, with more advanced versions on an advanced page. But I think nomenclature is a problem -- we shouldn't call such a cluster "basis (free module)", because the average person looking for the definition of the basis of a real vector space won't know what that means. On the other hand, calling the cluster "basis (vector space)" but including information about bases of free modules there is misleading. All of these could fall under "basis (mathematics)", but that page would be better as a disambiguation page for the uses of the word "basis" in all areas of mathematics.

I suggested above because it seems to me that "free module" is deserving of it's own page, and basis would have to be defined there anyway.Barry R. Smith 19:32, 27 November 2008 (UTC)

How about "Basis (linear algebra)"? Richard Pinch 20:00, 27 November 2008 (UTC)

Sounds good to me, as well as a page "free module", which doesn't seem to me to fall under what I would usually think of as linear algebra.Barry R. Smith 20:28, 27 November 2008 (UTC)

A basis is a minimal generating set. I think that covers all cases, doesn't it? That's a sufficiently well defined concept to be given in an article, if only a brief one. I don't think free module covers all cases. Topologies are modules, aren't they? But not free ones. What a generating set is depends on the operations concerned of course. Peter Jackson 11:56, 28 November 2008 (UTC)

Not entirely a rhetorical question. Long time since I did this stuff. Topologies aren't modules. Maybe semimodules? Certainly semirings. Anyway, that actually adds to the point. Free modules aren't the general case. The general case is my initial definition above. Peter Jackson 15:42, 28 November 2008 (UTC)

It's worth pointing out that in topology, while a base (more usual than basis) ofr a topology, or a filter base, may be regarded as a generating set, there is no implication of minimality. If a filter is principal, then its generator is not its basis. Topologies are very definitely not modules. Richard Pinch 18:14, 28 November 2008 (UTC)