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On an (personally, admittedly) cursory inspection, the section appeared only to deal with a definition of scalar multiplication which is better handled by the scalar multiplication article. Perhaps we can deal with a discussion of coordinate vectors and its relationships to "collections of scalars",
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I mostly reverted these two paragraphs to the original form. It seems inappropriate for a mathematics article to give priority to the informal and fuzzy sense of the term (which, by the way, was already listed further down in the head section) over the precise sense (which got largely lost in the
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I'd say your current introductory paragraph is considerably less useful than the ones you reverted and quoted above. It provides no insight at all as to the significance or rationale of the concept and is bound to be all but meaningless to anyone who is not a mathematician (to mathematicians it's
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In the section
Definitions and properties, subsection Scalars of vector spaces, second paragraph, the last part of the sentence appears to be ungrammatical and, thus, unintelligible: "The scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as
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As for getting rid of the "Scalars in vector spaces" section: for the above reasons, the first three head senses (at least) need to be expanded in the body of the article; and anyway there are some non-trivial things that can be said about the nature of the scalars. For instance, the additional
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constant. Also, it is not appropriate to say that a scalar "could be a nonvector element of a vector space", since "element of a vector space" is always understood to mean one of its vectors, never one of its scalars. Finally, there are six (at least) quite distinct senses in which the word
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Anyway, this line of discussion is appearing to move towards a rather extensive examination of these details, which really isn't warranted given the nature of the points of contention here. I'm not going to really pursue this further, since there are obviously better uses of both our time :)
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and other think the article was defining "scalars" as the coordinates of vectors. That would indeed have been utter nonsense, and Hardy's reaction was quite understandable. Since the purpose of the subsequent edits seemed to to be to patch up the defintion, but without removing that
728:. It may well be that the verb "to scale (up/down)" and the noun/adjective "scalar" evolved from the noun "scale" by separate routes: the former possibly among draftsmen and geometers, from the "scale of a drawing", the latter in algebra as per Hamilton's justification.
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As for the separate "Scalars in modules" section: I don't know if there is much else to say about the "scalars" of modules, other that they are just a ring. If that is the case, perhaps we can squeeze that material into a paragraph, and merge the two sections into
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You are right. The only fields I have ever seen used as scalars were either subfields of the complex numbers, or the integer numbers modulo a prime power. However, ther are weirder fields around (such as rational functions of a variable X).
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The rationale, if needed, should be added in the article body, as in other methematical articles. As for the style, and vector in modules, I have rewritten it again; please check. (BTW, what was wrong with "multiply into")?
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rewrite). Note that the informal sense is almost surely derived from the precise one (namely, 1x1 matrices are informally called scalars because they behave like elements of the scalar field), not the other way around.
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The introduction says : "...a scalar is a number..." I'm not very comfortable with that statement. A scalar need not be a number in any ususal (however general) sense of the word. I think it should be reworded somehow.
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OOps, that is not what I meant! The sense is: a vector space has three components (K,V,*) where K is a field (the scalars), V is an abelian group(?) (the vectors), and * is an operation from K x V to V satisfying...
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constraints that they must satisfy when the space has a inner product or a norm (see the invisible comment in the article). Also here one may add that, as a consequence of the basis theorem, every vector space (
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Granted the word "component" was a very bad choice. Perhaps "ingredient"? I do not think that one should mention "components" in the other sense, except perhaps in the context of the "coordinate space" example.
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Sorry, you are right, I owe an explanation. From the above comments and the subsequent edits, I realized that my use of the word "component" to describe the scalar field was very unfortuante, and it led
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is a thing composed of components, each of which is a scalar, is absurd, and to define "scalar" that way, in an article that should treat the concept in a more general way that that which appears at
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Particularly due to the encyclopedic nature of Wiki content, Mathematical articles should always possess an "Examples" section, for hard-examples or tangible definitions of the product in question.-
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merely a very poorly phrased tautology - do we multiply scalars "into" things? do we call the elements of modules "vectors"?). I'll leave it to other objective parties to compare the two, though.
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The suggestion that the "scalars" have that name because they "scale" (i.e. "change the scale") of vectors sounds plausible, but does not seem consistent with
Hamilton's quote; so it may be just
390:"scalar" occurs in mathematics, and they must all be listed in the head paragraph; so it seems best to keep each sense as succint as possible, to be amplified and elaborated in the article body.
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As for the changes to the head section: let me defend my version. The term "constant" has no absolute sense in mathematics, it is very context dependent; and anyway, in a scalar multiplication α
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Some people coming to this page are gonna not have a clue what "other elements of a field" are. Some common example of "other elements of a field" might be helpful. For example are integers.
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article, that tries to cover its uses in math, physics, and computing, and is pretty much along the lines of "simple value, without direction". I am assuming that readers who get to
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Hi. I've significantly refactored this article to improve organisation and tried to clarify some of the text, as well as simplify the intro per a suggestion from
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You also removed a lot of structural and generalizing edits as well. If you have no objection to these edits, I'd like to replace them.
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misunderstanding, it seemed best to revert and fix the real root of the problem. Sorry if that meant wasting the other editor's time.
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814:. If I inadvertently removed an important piece of information that someone added, I'd be grateful if you'd add it back in. Thanks!
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It is not sufficient to simply revert the page and state that you did so, Jorge. You are obliged to provide reasons for doing so.
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I have added a parag about every vector space being isomorphic to a coordinate space over its scalar field. Did I get it right?
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Yeah, constant isn't the best term, but if one can be found...? The introduction should aim to be as general as possible.
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Thus similarly shouldn't an indepth discussion of the mechanics of scalar multiplication be done at the
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The above is of course nonsense. A scalar may be a component of a vector, but to say that a
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Also, the details on what fields can be scalars are better left for the appropriate section.
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number in order to clearly distinguish such numbers from other numerical objects such as
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More generally, the set of scalars associated with a vector space may be any
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expect to see a more mathematically oriented (and hence precise) definition.
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There is already an informal and general "definition" of the term in the
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collections of scalars, after all—if one does not mind an isomorphism.
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I have reworked the head paragraph again. See if it is better now.
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Um, let me have another look at those edits. Hang on a minute...
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finite fields. a number by the elements inside the brackets."
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the notion of vector space is itself generalised by that of a
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As for moving the "Etymology" section up: fine with me (done).
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I didn't really touch that section, it looked OK to me.
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which would be a much more fruitful use of the article.
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Even though this article says scalars can be negative.
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Main sense should be the (precise) linear algebra sense
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Coordinate vector spaces are treated in the article
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it (changing its length), hence the choice of term.
754:I agree and hence I edited the intro slightly. --
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930:Knowledge level-5 vital articles in Mathematics
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