TITLE: Do all vectors belong to a vector space?
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: 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.