Can you define linear maps between vector spaces over different fields?

Question

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?

Answer 1

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.