Give a family of elements $\left(v_{1},\ldots,v_{n}\right)$ (where $v_{1},\ldots,v_{n}$ are just some sets), how many vector spaces are there, such that this family is a basis for that vector space ?
For example, if $n=1$ and $\left(v_{1}\right)=(1)$, there are at least two vector spaces, namely $\mathbb{Q}$ and $\mathbb{R}$ over themselves (as fields), such that this is a basis.
Is there even a reasonable way to answer this ?