Can you define linear maps between vector spaces over different fields? My book defines linear maps between vector spaces with a chosen fixed field but can you define linear maps between vector spaces over different fields? Is there any reason to restrict attention to fixed fields?

You could do something like this:
Suppose $V$ is a vector space over $F$ and $W$ is a vector space over $K$.
A linear transformation $T: V \to W$ in your sense would have to satisfy $T(\lambda v)= \lambda T(v)$ but the problem is that $\lambda\notin K$.
We can fix this by assuming a ring homomorphism $g: F \to K$ and then asking that $T(\lambda v)= g(\lambda) T(v)$.
This will work, but it is not anything more general:
Indeed, since $F$ is a field, $g$ must be injective. Therefore, $F$ can be seen as a subfield of $K$ and $W$ can be seen as a vector space over $F$, so there is nothing new.
