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On the history page for null vector at 15:57, 16 June 2008 Jitse Niesen said:
I like the addition of seminormed vector spaces, but the first sentence is now too complicated. How about starting with the easy case?
Yes, you are right. The first sentence was a bit much, but I still feel the need to point out, that the two kinds of null vectors considered are distinct. So I will try a second time, starting from your partial reversion. I hope that is fine with you.
Pilotkyber ( talk) 10:13, 17 June 2008 (UTC)
- Yes, that's fine. -- Jitse Niesen ( talk) 13:26, 17 June 2008 (UTC)
In Denmark we have a name ( da:Egentlig vektor translates directly "actual vector") for of all vectors that are not equal to the zero vector. Does there exist an equivalent in english? I was thinking that a "See also" section would be relevant in such a case. -- mgarde ( talk) 19:53, 29 March 2011 (UTC)
What things are in this article? First of all we have true zero vector, the additive identity element. It admits at least two generalizations:
- Null vectors in a seminormed vector space, i.e. v such as ‖v‖ = 0. This is a closed linear subspace.
- Null vectors in a
pseudoeuclidean space, i.e. v such as v∙v = 0. This is not a linear subspace but a quadric, and its linear span is the entire space. For example, is Minkowski space there exists a null basis:
So, there are two solutions.
- Split the article to 2 or 3 parts (zero vector, kernel subspace of a seminorm, and pseudoeuclidean null vectors which are very different from the former two), maybe with redirects to other articles. The name "null vector" may be occupied by already existing null vector (disambiguation) page.
- Make a vague lede, then three distinct sections with definitions and inbound redirects zero vector, null (seminorm) and null (pseudoeuclidean).
Any suggestions? Incnis Mrsi ( talk) 11:45, 29 February 2012 (UTC)
- I also found this article a little disconcerting, because it is not too clear on the distinct uses of the term. Splitting this into separate articles would place a heavy load on he disambiguation page, because the distinction is (to a non-mathematician) quite subtle: it requires using or defining some not necessarily obvious/familiar concepts. So I'd suggest modifying the article to make it clear that it is dealing with the various meanings of the term (your second option), and have a top-level section on each. The lead should then basically list the different uses of the term. Would the inbound redirects not be null vector (seminorm) and null vector (pseudoeuclidean)? – but are probably not needed anyway (without the disambiguator you get to the right article). The distinction between null and zero vectors must in any event be clearly made in the case of pseudoeuclidean spaces, as both are heavily used.
- It seems to me that the distinction between zero and null is the same in seminormed and pseudoeuclidean vector spaces (unless I'm misinterpreting something). The fact that the "subspace" (subset) of null vectors in the pseudoeuclidean case is not a linear space does not seem to me to be relevent to the distinction; in both cases the *-norm (pseudonorm/seminorm) determines what's null, and what's zero is equivalent in both cases. So the article could be written combining these cases, simply saying what properties are in each case. —
Quondum
☏
✎ 12:34, 29 February 2012 (UTC)
- There is an analogy, but not such one which would permit a universal generalized definition. A seminorm may be equal to the square root of a non-negative (but degenerate)
quadratic form, but not necessary is, just like not any norm is an Euclidean one, even in finite dimension. On the other hand, √v∙v (where ∙ is indefinite) by no means behaves like a norm, even in a such domain as the interior of a light cone where it is well defined. It is only a
homogeneous function on vectors, without any particularly nice properties on the whole space, and even not a real function, but a complex and
two-valued. The quadratic form v∙v itself is a homogeneous function of degree 2, not 1 as a norm must have. So, is null vector a "vector where some homogeneous function is equal to 0"?
Incnis Mrsi (
talk) 13:49, 29 February 2012 (UTC)
- I'll probably tie myself in knots making mathematical statements. I might have assumed that √|v∙v| would be adequately like a seminorm, except that the triangle inequality probably goes out of the window. Anyhow, I take your point: it'll be simpler to define a null vector in each type of space in its own section. The reader can then be left to notice the similarities if they wish. —
Quondum
☏
✎ 16:04, 29 February 2012 (UTC)
- The triangle inequality is wrong with √|∙| iff the quadratic form is strongly indefinite (has both + and − in its signature). I do not think that we should stuff a reader's mind with absolute values of quadratic forms and square roots, giving a function which is worth nothing. I propose to write that if the quadratic form has +s and 0s but not −s, then we obtain a special case of seminorm, but generally a quadratic form's nulls and a seminorm's nulls are different, although intersecting a little.
Incnis Mrsi (
talk) 16:22, 29 February 2012 (UTC)
- I agree, except there is probably more detail even with this than really needed. And how can you say a quadratic form's nulls and a seminorm's nulls can be different? I'd expect them to be the same if both are applicable for the same space. (Aside: √|∙| is not worthless: it is used for purposes of normalization in geometric algebra.) — Quondum ☏ ✎ 17:43, 29 February 2012 (UTC)
- The triangle inequality is wrong with √|∙| iff the quadratic form is strongly indefinite (has both + and − in its signature). I do not think that we should stuff a reader's mind with absolute values of quadratic forms and square roots, giving a function which is worth nothing. I propose to write that if the quadratic form has +s and 0s but not −s, then we obtain a special case of seminorm, but generally a quadratic form's nulls and a seminorm's nulls are different, although intersecting a little.
Incnis Mrsi (
talk) 16:22, 29 February 2012 (UTC)
- I'll probably tie myself in knots making mathematical statements. I might have assumed that √|v∙v| would be adequately like a seminorm, except that the triangle inequality probably goes out of the window. Anyhow, I take your point: it'll be simpler to define a null vector in each type of space in its own section. The reader can then be left to notice the similarities if they wish. —
Quondum
☏
✎ 16:04, 29 February 2012 (UTC)
- There is an analogy, but not such one which would permit a universal generalized definition. A seminorm may be equal to the square root of a non-negative (but degenerate)
quadratic form, but not necessary is, just like not any norm is an Euclidean one, even in finite dimension. On the other hand, √v∙v (where ∙ is indefinite) by no means behaves like a norm, even in a such domain as the interior of a light cone where it is well defined. It is only a
homogeneous function on vectors, without any particularly nice properties on the whole space, and even not a real function, but a complex and
two-valued. The quadratic form v∙v itself is a homogeneous function of degree 2, not 1 as a norm must have. So, is null vector a "vector where some homogeneous function is equal to 0"?
Incnis Mrsi (
talk) 13:49, 29 February 2012 (UTC)
- BTW, there exists an article additive identity. Should a reader of the zero vector link actually read all current stuff about seminorms and indefinite quadratic forms? Why would not redirect it to an explanation of the main property which distinguishes the true zero vector from various null generalizations? Incnis Mrsi ( talk) 18:49, 4 February 2013 (UTC)
This article has too much orientation to zero. For instance,
- In contexts in which the only null vector is the zero vector (such as Euclidean vector space) or where there is no defined concept of magnitude, null vector may be used as a synonym for zero vector.
Such statements require reference. The technical content of this article requires that it be considered in the correct context. The Shaw reference today explicitly states that a null vector is not zero. Rgdboer ( talk) 02:02, 11 March 2015 (UTC)
- I think we should take care not to give undue weight to Shaw with regard to the exclusion of the zero vector as a null vector; we should allow for the definition that defines it as a vector having zero magnitude.
- The sentence quoted above seem to be true often enough (there is no shortage of books that use it in this sense) that it is appropriate to mention this to avoid confusion, though we could demote this meaning to being a misnomer, maybe in a hatnote or footnote: we should at least retain the disambiguating effect. — Quondum 02:51, 11 March 2015 (UTC)
Since null vector has a precise meaning other than zero vector, the article has been rewritten, with references supplied. The old redirect of zero vector to this article has been changed to Zero element, with adjustment on that page. Rgdboer ( talk) 20:32, 17 March 2015 (UTC)
'seems to me that there should be a link to, say, " Euclidean vector" in there somewhere. For the reader who is curious about what a vector is. -- Jerome Potts ( talk) 15:41, 13 May 2015 (UTC)
- I've added a link to vector space, along with other changes. This should give the needed information. This article still needs lots of work. — Quondum 16:31, 13 May 2015 (UTC)
For a given linear transformation T or matrix M the kernel (linear algebra) is also known as the Null space of T or M. It seems that one might call a vector in the null space a null vector, which would be a meaning not in this article. This is a case of ambiguity that authors have already confronted, for instance the use of kernel instead of null space, but many older texts say null space. Furthermore, Thomas Cecil avoids the use of "null vector" by adopting the term lightlike vector in his book on Lie sphere geometry (page 10), referring to "lightlike subspace" when all vectors are null vectors(page 24). Question: should this potential ambiguity be raised in this article? Rgdboer ( talk) 20:56, 27 August 2015 (UTC)
To check this ambiguity, 10 textbooks in linear algebra were consulted, with one producing the anticipated result: Gilbert Strang (1980) second edition, page 75 says "sometimes a row vector yT is called a left nullvector of A" when yTA = 0. Though nullspace is the main term used, he also introduces kernel (page 72).
This survey of 10 found two texts that avoided null space and used only kernel: O’Mera/Clark/Vinsonhaler (2011) Advanced Topics, page 5, Akcoglu/Bartha/Ha (2009) Analysis in Vector Spaces, page 69.
Five texts used both terms: Strang (1980), Bloom (1979) Lineal Algebra and Geometry, page 245, Blyth/Robertson (1998) page 96. Two of these five introduce the two terms on separate pages: Johnson/Riess/Arnold third edition, p 187 null space, page 349 kernel. W.C. Brown (1991) p 122 null space, p 128 kernel.
Three texts have only null space: Shields (1980) p 366, Larson/Edwards (2000) p 216, Szidarovszky/Molnar (2001) p 265. With only the one source introducing the ambiguity, and a trend leading toward usage of kernel to avoid ambiguity, the suggestion may be suppressed here. Rgdboer ( talk) 22:10, 4 September 2015 (UTC)