Talk:Coordinate vector

Dumb Question

Shouldn't it be written this way,

$v=\alpha _{1}B_{1}+\alpha _{2}B_{2}+\cdots +\alpha _{n}B_{n}$,

i.e., With the $B_{i}$ all capitalized? They are vectors after all.192.55.12.36 (talk) 21:39, 22 April 2008 (UTC)

The names of vectors are not normally capitalised in my experience. Sometimes (generally in applied mathematics) they are bolded or underlined, but often they're just in lowercase italics, as here. Algebraist 17:18, 25 April 2008 (UTC)

Definition

I read this page while trying to gain an understanding of what a coordinate vector was, and the initial definition really did not help me; it uses the word vector in it which is what I'm trying to define, e.g. it is partly read like this: 'A coordinate vector is... ...a vector...' This does not help me because there is no link on the word vector and typing it in just brings me to a disambiguation page.Dood77 (talk) 15:39, 28 November 2007 (UTC)

I think you are confused because the article uses "vector" to mean two different things: Sometimes it means a member of a vector space, and sometimes it means a matrix with just one column or just one row. I think the article could be changed to make the different meanings more obvious, but I don't think that I am the right person to fix it; I'm too much of a tyro.192.55.12.36 (talk) 20:16, 25 April 2008 (UTC)

naming/merge

I'm unilaterally renaming this page from "coordinates vector" to "coordinate vector", which makes better sense and seems to be the consensus on Google. Melchoir 05:02, 10 November 2005 (UTC)

Perhaps it should be merged with coordinate space? Jorge Stolfi 12:57, 4 January 2006 (UTC)

Yeah, I'd support that merge, although I wouldn't want to do it myself! The Discussion section of Coordinate space is a good target. Melchoir 05:34, 5 January 2006 (UTC)

I'm not sure a merge there would be a good idea. I would rather see this page moved to coordinate representation and have it dicuss coordinate representations of vectors, dual-vectors, and linear transformations in a unified manner. It could then discuss coordinate transformations which would just look confusing in the page on coordinate space. The treatement in the coordinate space article should serve to complement what should be here. -- Fropuff 06:40, 5 January 2006 (UTC)

Or maybe row vector and column vector should be merged into coordinate vector. Duoduoduo (talk) 18:49, 12 December 2011 (UTC)

Questionable restriction in the definition

1. A coordinate vector is not necessarily a matrix. In this case, you can use the notation v = (v₁, v₂, ..., v_n), in which the coordinates are not supposed to be elements of a matrix, but just a generic ordered set.
2. When you define it as a matrix, it must not necessarily be a column matrix (column vector).

I have seen many people (in computer graphics, for instance) who prefer using row vectors (and this entails post-multiplication by basis transformation matrices, rather than pre-multiplication). Thus, there are two different and equally valid conventions, referred to as "column vector" and "row vector" conventions. None of them can be shown to be objectiely preferable. I am not even sure that the first is more commonly used than the second. Paolo.dL (talk) 09:37, 4 June 2008 (UTC)

Both this page and "coordinate space" are confusing and nonstandard

The definition given in "coordinate space" is bizarre: what is the "product [topological] space of a field over a set," given that, a priori, a field has no topology? In any event, why drag topology into the discussion at all? For that matter, why require the index set to be finite? What the author seems to have in mind is the free vector space over F generated by some given index set I, i.e., "finite formal linear combinations of elements in I with coefficients in F."

Next, the page's claim that, given an ordered basis, each vector has a unique representation as a [finite] linear combination of vectors is confusing, since ordering has nothing to do with it (or anything else in linear algebra other than describing particular means of calculation) — vector addition is, after all, commutative. Also, the coefficients aren't really determined uniquely by "one of the defining properties of bases" unless one takes the entire definition to be "one property": a basis is a linearly independent spanning set. If a set B does not span the space, there exists a vector that cannot be represented as a finite linear combination of elements from B, and if B is not linearly independent, the expression is not unique, since some finite linear combination of vectors in B with nonzero coefficients is itself a vector in B. There are no other defining properties.

"Typically, but not necessarily, the coordinates are represented as elements of a column vector." Maybe so — in certain contexts. In others, they're typically represented as row vectors, or tuples, and in others, the term "coordinate" is taken to mean a linear map from vectors to numbers (also called "coordinates") — and the "coordinates," while elements of a vector space, are elements of a different vector space from the vector space V they coordinatize, namely, the dual space V^*. Incidentally, given any "reasonable" application of the word "coordinate," there exists a differential geometry book that uses it in this way (or so it seems), so perhaps all references to "coordinates" in mathematics articles should lead to a very long "disambiguation" page. — Preceding unsigned comment added by 75.184.125.61 (talk) 09:47, 1 June 2011 (UTC)

The need for an ordered basis is quite important. The coordinate vector of an abstract vector is the ordered n-tuple consisting of coefficients of the linear combination representing it. You're right, vector addition is commutative, so that 2*[1,0]+3*[0,1]=3*[0,1]+2*[1,0], but the coordinate vector will be [2,3] or [3,2] depending on which order is chosen for the basis [1,0] and [0,1]. This order on the basis allows for unique representation of coordinates. Rschwieb (talk) 14:18, 18 October 2011 (UTC)

Expert

See Talk:Coordinate space for comments on expert help. D O N D E groovily Talk to me 04:42, 18 October 2011 (UTC)