Does zero vector only exist in dim 0?

[pob1212]

Main Question or Discussion Point

Hi,

just curious, is it false to say the vector 0 exists in R^n, n>0 and a natural number?
i.e. does x = [0, 0, 0, 0] exist in R^4, or simply the zero space?
My guess is that the first statement is not false given that the zero space is a subspace of R^n, n>0.

Am I right or wrong?

Thanks

 

[Number Nine]

The zero vector is the additive identity; it exists in all Rⁿ, and is defined as $<0, 0, ... , 0>$.

EDIT: My statement is misleading. "The zero vector" does not exist in every Rⁿ; rather, for every n, there exists an element of Rⁿ which is the additive identity and is called the zero vector.

 

[HallsofIvy]

But in what sense is the vector "which is the additive identity and is called the zero vector" NOT the "zero vector"?

But, having written that, I do see your point. It is a mistake to think that there exists a single vector that is in all vector spaces. In, for example, R³, the "zero vector" is (0, 0, 0). But in the vector space of "all polynomials of degree 2 or less" the "zero vector" is the function f(x) such that f(x)= 0.

Every vector space contains a zero vector but it is misleading to talk about "the" zero vector as if all zero vectors were the same thing.

(But all vector spaces of dimension n are isomorphic and that isomorphism maps the zero vector in one to the zero vector in the other. A "natural isomorphism" from R³ to the space of polynomials of degree 2 or less takes (a, b, c) to ax^2+ bx+ c and so maps (0, 0, 0) to 0x^2+ 0x+ 0= 0 for all x.)