Talk:Null vector

Two kinds of null vectors

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)

Vector that is not the zero vector

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)

Either split, or make distinct sections

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:

  $\begin{pmatrix}1\\1\\0\\0\end{pmatrix} \begin{pmatrix}1\\0\\1\\0\end{pmatrix} \begin{pmatrix}1\\0\\0\\1\end{pmatrix} \begin{pmatrix}1\\0\\0\\-1\end{pmatrix}$

So, there are two solutions.

1. 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.
2. 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)