996:
of definition of "bilinear" is a problem. I'm not really sure what to do here; if one defines "bilinear" then one should give an example, but the simplest example is the dot product, which is later in the article. And being vague, as Stca74 noted, is no solution either. It might be good to introduce the dot product here and then reintroduce it later in the inner product section; the second time you'd point out that it's positive definite. (Also, the inner product section currently calls the
Minkowski form an "inner product" even though it's not positive definite. I know that in physics, "inner product" doesn't mean positive definite, but it certainly does in math. This deserves a remark somewhere, I think.)
932:. Then one could state that a space with these properties does exits, and outline the construction. Similarly the adjoint property of tensor product with respect to Hom is too vague. To control article size, one could consider leaving that out as tensor product article is wikilinked; otherwise one should definitely point out that tensor product is a (bi-) functor. As for extension (and restriction) of scalars (tensoring with extension field of the base field), that could be treated, but then again functoriality of tensor product would be natural to include. Perhaps effective use of summary style could help keep amount of material here still manageable.
1036:
best to have a section on bilinear forms first (which would mention inner products and the
Minkowski metric and link to the article on signature) and then use those to justify tensor products: "Tensor products let us talk about bilinear maps, which you now know to be wonderful, in terms of linear maps, which you also know to be wonderful." That would require reorganizing the article a little, but I don't see a good other solution.
963:
representation of Hom as tensor product of dual of the domain with the target, I'm not sure if I can follow (or expect others to follow). Actually, the canonical map goes in general from the tensor product into the Hom space and is injective. It is bijective if one of the spaces is finite dimensional. Thus, if you insist, you get an interpretation of a tensor (element of the tensor product) as a matrix, but not really tensor as a
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1104:) and thus there are no interesting norms to put on vector spaces over finite fields. And while norms on finite-dimensional real vector spaces equivalent, there are still no canonical norms nor inner products. I do agree with Ozob's view that it makes sense to warn readers about this potentially counterintuitive fact.
948:
deferred to the subpage. As for the isomorphism: I don't know why I called this adjunction isomorphism, since it is effectively both adjunction and reflexivity of f.d. spaces. Anyhow, this comment was just to put tensors in line with scalars, vectors and matrices, but I would not go into functoriality etc.
627:
That's exactly what I was thinking of. I don't want to make a big deal about that construction, but I do think it's good to mention—it's the right way to think about the determinant, and the only way I can think of which admits generalizations (e.g. to vector bundles). I put a sentence in the article
1035:
Oof, that's a tough way of figuring it out! (Not that I did better!) I agree, this is a tough thing to work out. It'll have to be done before FA, though (if that's where you want to take the article). The only really elementary context I can think of where they turn up is bilinear forms. It might be
995:
Unfortunately, the tensor product section now has a problem: It doesn't define "bilinear", so it doesn't make a lot of sense. The previous version was better in this respect because it was only hand-waving, so the reader didn't expect to understand; but now that the article is more precise, the lack
947:
Thank you too, Stca, for your review: I have trimmed down the tensor product discussion a bit, but also made it more concrete. I think doing the universal property thing properly (i.e. with explanation) is too long and also a bit too complicated (even uninteresting?) for general folks, so should be
1065:
Maybe you should not emphasize this (nor should you write that they do have these structures) because you can equip vector spaces with a norm (distance and nearness) or an inner product (for angles) and I am quite sure that most of mathematics done on vector spaces studies these structures on them
414:
I'm not sure I understand your points. I do mean z to be complex, just for concreteness. ("Another example is Q(z), the smallest field containing the rationals and some complex number z.") What is the problem with z=1 and a trivial extension? What do you mean by "it's not unique"? (I think, for
962:
Looks more precise now. However, I would still consider adding the universal property (perhaps somewhat informally, at least) - as least for me it is the only way to make sense of the construction, which otherwise risks being just a tangle of formulas. As for the last few lines after the
635:
It seems that for most of the article, whenever you need an example of a non-abstract vector space, you use solutions to differential equations. I agree wholeheartedly that these are important, but there are probably other good examples out there which shouldn't be slighted.
469:
The bolded expression〈x | y〉does not display properly on Safari 3.0.4; the left and right hand angle brackets show up as squares, Safari's usual notation for "I don't have this character". (It works when unbolded, as I found out when I previewed this page.)
1066:(such as Banach space theory or Riemannian geometry). So perhaps keeping the sentence, should imply that there is an explanation that you still can equip these structures on vector spaces since these strucutures are indeed very important in mathematics.
665:
It also seems that you rely on convergence to justify the introduction of other structures such inner products; but inner products can be (and should be, I think) justified on geometric terms, because they're necessary to define the notion of an angle.
359:
The lead says "much of their theory is of a linear nature"; but I don't think the meaning of "linear nature" will be apparent to people unfamiliar with vector spaces. E.g., someone might not know what a linear combination or linear transformation is.
1020:
subsection that is inviting enough to guide the reader to subarticle. When I learnt this, I kept wondering "what is this u.pr. all about?" I only got it after learning about fiber products of (affine) schemes, but we certainly cannot put that up!
373:
In the "Motivation and definition" and definition section, "linear combination" has not yet been defined. It might be better to say that there is no preferred set of numbers for a vector, and to say no more until bases have been introduced.
584:
Well, I'm not sure what's the best way to state it. But I feel like the fact that all finite-dimensional spaces are reflexive is really, really important and needs to be mentioned somewhere. The way you have it now is fine.
530:(p. 28, exercise 9). But it seems to me that the best reason for discussing them is the finite-dimensional case: Right now, the article doesn't discuss reflexivity of finite-dimensional vector spaces, a real gap!
612:
I'm not sure that likening a basis for a vector space to generators for a group or a basis for a topology will help most readers. Most people who use and need linear algebra have never heard of these.
624:
Except for det (f: V → V) being related to Λ f: Λ V → Λ V (which I think should not be touched here), I don't see why the determinant belongs to multilinear algebra. What specifically do you think of?
512:
I now had a look at most of the algebra books listed in the article and none of them, actually, talks about algebraic biduals. So I wonder if this is so important. (I wondered already before).
407:=1 then the field extension is trivial; even if the field extension is non-trivial, it's not unique (square root of 2 vs. cube root of 2). I see below that you do really mean for
786:
880:
The description of tensor product as it stands is too vague (such as "mimicking bilinearity"). It would be better to first give the universal property of tensor product of
738:
483:
The natural map V → V** is only discussed in the topological setting. It should be discussed in general. (Note that the map is always injective if one considers the
461:
The article should say very early that abstract vector spaces don't have a notion of an angle or of distance or of nearness. This is confusing for most people.
1061:
The article should say very early that abstract vector spaces don't have a notion of an angle or of distance or of nearness. This is confusing for most people.
877:
I concur with the comments above; I have the following comment to make on tensor products, which I would like to be taken addressed before GA status:
621:
Since you mention the determinant, it's worth mentioning that it's a construction from multilinear algebra. A sentence or two should suffice.
66:
570:
What else do you want ("... This is in contrast to the linear-algebraic bidual, i.e. where no continuity requirements are imposed:... ")?
91:
85:
62:
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47:
473:
Yeah, it was weird, there were two types of angle brackets. Can you read them now (there are three occurences in that section).
170:
434:
looks like a transcendental to me. This might have been part of my confusion, too. But change it back if you think having a
39:
806:), but somehow the strip (which was taken from a photo, so png previously) is invisible to me?! Any ideas about that?
556:
I'm almost ready to say this is a GA; my only outstanding issue is reflexivity of finite dimensional vector spaces.
1016:(<-) Well, the bilinearity is certainly no problem. I mentioned this now. The problem is more: how to create a
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simplicity, the subfield-of-C-definition I'm giving is appropriate at this stage, and yields something unique).
426:(plural). I've changed the text to try to make this better; is this OK for you? (BTW, I used an α instead of a
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OK, I scrapped the sketched ladder of "tensority". Also the universal property should be fine now.
608:
Here are some other issues which aren't as pressing but which I think you should handle before FA:
751:
705:
55:
17:
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972:
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441:α is fine, but I will have to eliminate the redundancy with the section on dimension later.
269:
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639:
Hm, I also talk about (non-differential) equations, but what else do you have in mind?
590:
561:
343:
218:
32:
1053:
I will try to review each section one by one and add comments. But just something
1096:
Careful with generalisations: any absolute value on a finite field is improper (|
1105:
968:
933:
678:
It's also worth mentioning the use of vector spaces in representation theory.
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355:
Here are some specific issues that I'd like fixed before this reaches GA:
347:
539:
JPEG uses a discrete cosine transform, not a discrete
Fourier transform.
418:
I think what bothers me is that you say you are about to give another
411:
to be complex, but perhaps there's a better way to say what you mean.
684:
Good, that's as much on representation theory as this article needs.
642:
I'm not sure! I was hoping someone else here would have good ideas.
828:
I'll try to make a SVG picture of a moebius strip later today. (
740:, the output looks better if you put a thinspace (a \,) between
387:
In the subheading "Field extensions", the description of
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967:of matrix (following scalar, vector, matrix list).
780:
732:
422:(singular) and then proceed to give a family of
904:with the universal property of expressing all
288:lack of images (does not in itself exclude GA)
8:
273:, where possible, to illustrate the topic.
802:I tried to convert the image into an svg (
491:, use the linear functional "project onto
862:Thanks very much, Ozob, for your review!
753:
707:
526:Hmm! I know that they appear in Halmos's
399:to be a transcendental, but you say that
770:
395:) is odd: It sounds like you mean for
7:
616:I'm not sure either! I removed it.
498:Done. (will provide a ref. later)
24:
702:When writing an integral such as
528:Finite dimensional vector spaces
464:OK. (In the definition section).
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159:(citations to reliable sources)
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1:
1046:03:45, 12 December 2008 (UTC)
1031:08:07, 10 December 2008 (UTC)
1006:00:55, 10 December 2008 (UTC)
781:{\displaystyle \int f(x)\,dx}
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595:03:36, 12 December 2008 (UTC)
580:15:53, 10 December 2008 (UTC)
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348:03:33, 12 December 2008 (UTC)
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991:21:40, 9 December 2008 (UTC)
977:20:06, 7 December 2008 (UTC)
958:15:12, 7 December 2008 (UTC)
942:10:45, 7 December 2008 (UTC)
872:12:04, 7 December 2008 (UTC)
857:02:40, 7 December 2008 (UTC)
838:10:58, 8 December 2008 (UTC)
816:12:50, 7 December 2008 (UTC)
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522:21:40, 9 December 2008 (UTC)
508:12:50, 7 December 2008 (UTC)
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888:as the unique vector space
733:{\displaystyle \int f(x)dx}
338:You pass. Congratulations!
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238:(all significant views)
101:reasonably well written
804:Image:Moebiusstrip.svg
797:Image:Moebiusstrip.png
782:
734:
681:Done. (Very briefly).
298:(non-free images have
278:(tagged and captioned)
1100:|=1 for all non-zero
783:
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228:(fair representation)
219:neutral point of view
187:broad in its coverage
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300:fair use rationales
916:to a vector space
799:should be an SVG.
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136:factually accurate
671:OK, you did this.
18:Talk:Vector space
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403:is complex. If
363:OK. Better now?
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1083:Topology Expert
1070:Topology Expert
1023:Jakob.scholbach
983:Jakob.scholbach
950:Jakob.scholbach
892:+ bilinear map
864:Jakob.scholbach
808:Jakob.scholbach
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647:Jakob.scholbach
572:Jakob.scholbach
544:Jakob.scholbach
514:Jakob.scholbach
500:Jakob.scholbach
487:dual (for each
443:Jakob.scholbach
216:It follows the
194:(major aspects)
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645:All right ;)
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438:is better.)
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57:Article talk
56:
52:
33:
30:
830:TimothyRias
628:about this.
44:visual edit
924:maps from
908:maps from
380:Also good.
141:verifiable
1055:User:Ozob
485:algebraic
322:Pass/Fail
268:contains
204:(focused)
27:GA Review
906:bilinear
430:because
424:examples
420:example
315:Overall
108:(prose)
67:history
48:history
34:Article
1106:Stca74
1057:said:
1018:little
969:Stca74
934:Stca74
922:linear
270:images
252:stable
250:It is
221:policy
185:It is
134:It is
99:It is
88:review
920:as a
495:".))
90:(see
76:Watch
16:<
1110:talk
1087:talk
1074:talk
1042:talk
1038:Ozob
1027:talk
1002:talk
998:Ozob
987:talk
973:talk
954:talk
938:talk
884:and
868:talk
853:talk
849:Ozob
834:talk
812:talk
744:and
690:talk
686:Ozob
651:talk
591:talk
587:Ozob
576:talk
562:talk
558:Ozob
548:talk
542:OK.
518:talk
504:talk
476:Yes.
447:talk
377:OK.
366:Yes.
344:talk
340:Ozob
138:and
92:here
63:edit
40:edit
928:to
791:OK.
266:It
120:MoS
1112:)
1089:)
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975:)
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940:)
912:×
900:→
896:×
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788:.
756:∫
748::
746:dx
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171:OR
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110::
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86:GA
65:|
46:|
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