- #1

Rasalhague

- 1,387

- 2

"A vector [...] is a combination of direction and magnitude. Vectors have no other properties: the magnitude and direction specify a vector completely" (D.H. Griffel:

*Linear Algebra and its Applications*, Volume 1, p. 3),

and (2) algebraic, e.g.

A vector is element of a set

*V*associationed with a field, [itex]\mathbb{K}[/itex], and operations called vector addition, [itex]V \times V \to V[/itex], and scalar multiplication, [itex]\mathbb{K} \times V \to V[/itex], such as to comprise a vector space, according to the vector space axioms.

Is the algebraic definition more general, and the geometric definition a special case of the algebraic, or are they in some sense equivalent (perhaps given some generalised, abstract definition of direction)?

My second question is about the relationship of vectors to their components and the effects of a change of basis. A lot of the texts I've been reading have emphasised that a vector (at least when considered as a type-(1,0) tensor) is invariant under a change of basis. I take this to mean that the vector itself is thought of, in this sense, as existing independently of its coordinate representation, and that there are infinitely many ordered sets of

*n*numbers which could represent the same n-vector (there being just one such set for each basis), and infinitely many n-vectors which can be represented by a given ordered set of

*n*numbers (there being just one such n-vector for each basis). I was thinking of "n-vector" as meaning an element of an n-dimensional vector space, a vector which requires

*n*components, regardless of what those components happen to be in any particular basis.

And yet I also read statements like, "For any positive integer,

*n*, and

*is an ordered set of*

**n-vector***n*numbers" (Griffel, p. 22). And in Chapter 3, e.g. p. 77, Griffel uses the term n-vector for an element of a vector space [itex]\mathbb{K}^{n}[/itex] over a field [itex]\mathbb{K}[/itex]. These n-vectors are presumably ordered sets of

*n*elelements, each element of which is an element of [itex]\mathbb{K}[/itex].

So, if "every n-dimensional [vector] space over [itex]\mathbb{K}[/itex] is isomorphic to [itex]\mathbb{K}^{n}[/itex]" (Griffel, p. 77), an isomorphism being given by each basis, does this imply that [itex]\mathbb{K}^{n}[/itex] can only have one basis? The name "standard basis" suggests not, but if we allow the possibility of a change of basis in [itex]J : \mathbb{K}^{n} \to \mathbb{K}^{n}[/itex], and if elements of [itex]\mathbb{K}^{n}[/itex] are not defined geometrically, by magnitude and direction, but rather defined (as Griffel defines them) as ordered sets of

*n*particular numbers, then the change of basis may transform one n-vector into a completely different n-vector, rather than simply relabelling it with a different set of components, and would therefore not be a change of basis in the traditional sense, or would it?

Or should I be thinking of a change of basis for [itex]\mathbb{K}^{n}[/itex] as involving multiple copies of [itex]\mathbb{K}^{n}[/itex], one in which the vector itself exists in some sort of absolute sense, and say two other copies of [itex]\mathbb{K}^{n}[/itex] playing the role of coordinate systems in which the vector itself is merely represented (by the same or a different arbitrary element of [itex]\mathbb{K}^{n}[/itex])? I'm thinking here of the definitions of a manifold that I've read and am still digesting. If I've got this right, the vector space [itex]\mathbb{K}^{n}[/itex] where the vector itself lives, in this absolute sense, would be the manifold, and the two other copies, the domain and range of the change of basis [itex]J : \mathbb{K}^{n} \to \mathbb{K}^{n}[/itex], would be mere placeholders or arbitrary ways of representing/incarnating the vector as a set of components. (This viewpoint presumably coming into its own when the manifold is something more complicated than [itex]\mathbb{K}^{n}[/itex].)