Talk:Basis (linear algebra)

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1 Hamel bases
2 Ordered Bases
3 Request for technical explanation
4 Too Technical
5 New To Advanced Math
6 Co-ordinate vs. Coordinate
7 Definition
8 Is "By brute force" a colloquialism?
9 Unclear wording about Hamel basis
10 Shauder dimension

[edit] Hamel bases

The page Hamel basis redirects here, and I'm not sure that is appropriate. If it is appropriate, important information about Hamel bases is missing from this article. Hamel bases are discussed most frequently as bases for the real numbers considered as a vector space over the rationals, and the article here omits that important idea. I'm going to remove the redirection and add "Hamel basis" to the list of requested mathematics article, and possibly supply a new "Hamel basis" article myself when I can. -- Dominus 18:46, 20 May 2004 (UTC)

You have a point. But in the scheme of things I'd say the point made at length about orthonormal bases not being vector space bases is a more central topic.

A dedicated article about Hamel bases for R over Q would be fine, IMO. There is enough to say.

By the way, the introduction of the idea of basis by the four conditions that can be proved equivalent is a bad old Bourbakiste trick, unsuitable for WP. Basis od a vector space is something very fundamental in algebra. It needs a more gentle and readable introduction. Charles Matthews 12:40, 6 Mar 2005 (UTC)

I'm curious about the following statement: Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are of little if any interest. I have the following problem in mind: let V be the space of infinite sequences of reals with finitely many nonzero components. The standard Hamel basis of this space is countably infinite. To prove that the dual of V, V*, is not isomorphic to V, one might show that any Hamel basis of V* is uncountable, no? 66.235.51.96 03:51, 12 November 2005 (UTC)

[edit] Ordered Bases

More often than not bases are ordered. None of the examples uses set notation for the basis, all of them present an ordered basis. If nobody vetos, I will change the definition to "family of vectors". As any set X is naturally a family (x)_x∈X the current definition is included in the new one. Markus Schmaus 21:24, 14 Jun 2005 (UTC)

Of course, elements in a set may not be ordered, and since a basis is a set, the basis may not be ordered. So I think I am not sure why we have to say a basis is a family of vectors not a set. Probably putting some example of non-ordered bases to the article is helpful for both us and readers. -- Taku 22:11, Jun 14, 2005 (UTC)

Is ( (1,0), (1,0), (0,1) ) a basis of R²? Why? Markus Schmaus 15:09, 15 Jun 2005 (UTC)

No, a basis is a set and ( (1,0), (1,0), (0,1) ) is, at least in the notation that I use, not a set. I don't see where in the examples in the article the ordering is used. Do you have any references which define the basis as a family? All references I checked define it as a set. -- Jitse Niesen 16:14, 15 Jun 2005 (UTC)

I am not sure what you mean by ( (1,0), (1,0), (0,1) ). Is it like, let a be a pair (1, 0) and b (0, 1). Then { (a, a, b) } forms a basis of R²? Isn't that a b forms such a basis? -- Taku 21:12, Jun 15, 2005 (UTC)

[edit] Request for technical explanation

This article is crying out for a picture and an example using R³ or R² with analogy to a simple XYZ or XY coordinate system. --Beland 16:17, 18 December 2005 (UTC)

[edit] Too Technical

I am trying to understand dirac notation by reading wikipedia, and I am finding that all the articles are very technical. Now, "basis" is an idea I can wrap my head around, so I'll add an informal thingo. If you find it inaccurate, tweak it by all means but don't just remove purely it on the basis that "this is so basic that everyone already knows it".

[edit] New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as basis, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

Hmmm, in my experience learning university-level mathematics was a fairly involved project that would have been difficult to do off the web :). I highly recommend finding a university and at least sitting in on their classes (or better, enrolling), if you possibly can. -- pde 23:19, 8 March 2006 (UTC)

[edit] Co-ordinate vs. Coordinate

Assuming there's a difference, can someone explain why there would be inconsistent references in this article? It would only make sense logically to use one and use the same throughout (e.g. don't call apples oranges, and vice versa--just call it what you normally call it). -therearenospoons 16:30, 12 April 2006 (UTC)

[edit] Definition

Recently, the definition was changed from:

"Let B be a subset of a vector space V. A linear combination is a finite sum of the form

$a_1 v_1 + \cdots + a_n v_n, \,$

where the v_k are different vectors from B and the a_k are scalars. The vectors in B are linearly independent if the only linear combinations adding up to the zero vector have a₁ = ... = a_n = 0. The set B is a generating set if every vector in V is a linear combination of vectors in B. Finally, B is a basis if it is a generating set of linearly independent vectors."

to:

"A basis B of a vector space V is a linearly independant subset of V that spans (or generates) V.

If B is endowed with a specific order (i.e. B has a finite sequence of linearly independant vectors that generate V), then B is called an ordered basis."

The editor said that this simplifies the definition. While the second definition is definitely shorter, it does not explain what "linearly independent" and "spans" means. For that reason, I prefer the first definition. I expect that most people who know what "linearly independent" and "spans" means, also know what "basis" means and do not need to read the definition. -- Jitse Niesen (talk) 04:57, 28 April 2006 (UTC)

I added some more detail in the definition, however, the definition of a basis is strictly what is in the first line (reference: Linear Algebra, 4th edition, Friedberg, Insel, Spence). The extra info is just an explanation of what the definition means. --Spindled 05:41, 28 April 2006 (UTC)

I like that the new section starts with the most important fact, what a basis is. However, I did make some edits to it (covered the infinite case, there is no A in independent in English, only variables should be italics but no parentheses, numbers, or other symbols); I hope you don't mind. By the way, have you already found Wikipedia:WikiProject Mathematics? -- Jitse Niesen (talk) 12:18, 28 April 2006 (UTC)

[edit] Is "By brute force" a colloquialism?

The example of alternative proofs is terrific. However, the phrase "By brute force" sounds like a colloquialism, so I have changed it to "By algebra". I considered "By algebraic manipulation", which sounds like something a con artist would do, and "By calculation", which sounds like something a calculator could do. :-)

I have removed the characterization of alternative proof methods because it implied that anyone who uses the first method is unsophisticated. That is POV. Perhaps someone can rewrite it. --Jtir 18:58, 26 September 2006 (UTC)

Personally, I don't think 'brute force' is too colloquial, but on the other hand I on't think it is the right phrase to use in this case. The proof of independence is simply going back to the definition of independence, so maybe ... 'from the definition' might be appropriate? Madmath789 19:33, 26 September 2006 (UTC)

That's much better! I would never have thought of that because I was hung up on maintaining the "By ..." pattern. I have made the edit, which includes a smoother intro. --Jtir 20:16, 26 September 2006 (UTC)

[edit] Unclear wording about Hamel basis

The section titled Hamel Basis currently begins with the sentance:

The phrase Hamel basis is sometimes used to refer to a basis as defined above, in which the fact that all linear combinations are finite is crucial.

But what lies above is the definition for the ordered basis. Is this sentence implying the Hamel basis is a kind of ordered basis? If so, then this should be stated explicitly, instead of the vague "defined above" referent. The second part of the sentence, ...is crucial is also confusing: the sections above don't really discuss finiteness, and so insisting that finiteness is now crucial is even more confusing. Can someone please fix this up? linas 23:20, 14 October 2006 (UTC)

Is it better now or just more confusing? -- Jitse Niesen (talk) 03:26, 15 October 2006 (UTC)

[edit] Shauder dimension

I'm not sure, but it seems to me by knowledge that Shauder basis have not to be of the same cardinality. Consequently Shauder dimension should be defined more clearly. —The preceding unsigned comment was added by 195.220.60.4 (talk) 12:08, 15 December 2006 (UTC).

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