TITLE: Why do we call a vector space in terms of vector space over some field?
QUESTION [9 upvotes]: I am getting a bit confused with the terminology here. I understand that a field means some set of scalars like real numbers but why do we need a field for a vector space? Are not the numerical values used to define a vector are inherent properties of the vector space? Why do we term it like "vector space over real field"? If not numbers, what are the other fields possible for a vector space because you obviously numbers to define the value of magnitude and direction of the vectors?

REPLY [2 votes]: Are not the numerical values used to define a vector are inherent properties of the vector space?

Yes, they are. If you use other numerical values (like complex numbers, or $\mathbb F_2$), you get another vector space (for example $\mathbb F_2^3$ instead of $\mathbb R^3$). However you can also get a different vector spaces using the same field (for example, $\mathbb R^2$ and $\mathbb R^3$). Now it turns out that vector spaces over the same field are in some sense compatible with each other, in a way that vector spaces over different fields generally are not. For example, it's no problem to define a linear function $f:\mathbb R^2\to\mathbb R^3$. However there are no linear functions $f:\mathbb F_2^2\to\mathbb R^2$. Therefore it makes sense to always explicitly say on which field the vector space is based.