Do all vectors belong to a vector space? If we were given the vector $(1,1,1)$, say, we know immediately that it belongs to the vector space $\mathbb{R}^3$ (and infinitely many others).  But, if we take this vector of dolphins:

or some other bizarre contrived vector: Does it belong to a vector space?
The vector Wikipedia page seems to define them as:

An element of a vector space.

This suggests the answer to the above question is yes; if they weren't in a vector space, they wouldn't be vectors.  But I'm not convinced by this sentence alone.
 A: A vector space always exists over a certain field. In the case of your dolphins, for it to be part of a vector space, two things would have to happen:


*

*You would have to define the $+$ and $\times$ operators for the field of dolphins, define a "zero dolphin", and prove that dolphins abide by all of the field axioms.

*You would have show that the set of all dolphin vectors is closed on vector addition, and scalar multiplication.
Otherwise, it is not a vector, but an ordered set.
A: If a vector belongs to a vector space, its components must belong to a field $K$.
This is not the case for dolphins. A similar question is, even using numers instead of dolphins: is the set $\lbrace 2 \rbrace \subset \mathbb{R}$ on the real line a real vector space ? But then it should contain $\lbrace 0\rbrace$, which it does not. On the other hand, can we define $2$ as the zero element ?
A: A vector, by definition, is an element of vector space, which is a set of vectors satisfying eight axioms. So no matter what the element is, even dolphin, it is a vector if and only if all elements in consideration form a vector space.
For example, which is the simplest, the "dolphin" is a vector if the vector space on $\mathcal{Z}_2$ is $\{0,''dolphin''\}$ and $''dolphin''+''dolphin''=0$
