The title pretty much says it all; is a vector space meaningless without a choice of basis which has been predefined? I've been reading through my algebra book and it says something like, take a vector $v$ and let $x$ be its coordinate with respect to a particular basis. My question is, how can you even take a vector before you assign a basis to its vector space? I know similar questions may have been asked here, but they all involve something about the Axiom of Completeness, which I have not learned about yet. Thanks so much for your help!
Any set of objects that satisfies the relevant axioms is a vector space. Not all vectors are tuples of scalars, although that’s certainly what you work with most in introductory courses. For instance, the set of real-valued functions defined on the interval $[0,1]$ forms a vector space over the reals. I’d be hard-pressed to even define a basis for this space at all.
Now, for a finite-dimensional space over some field $\mathbb K$, choosing a basis amounts to defining an isomorphism between the space and $\mathbb K^n$, which is why you can focus on tuples of scalars when studying finite-dimensional vector spaces.