TITLE: ''Linear'' transformations between vector spaces over different fields .
QUESTION [7 upvotes]: Let $\mathbf{V}(\mathbb{K}_1,V),$ and $\mathbf{W}(\mathbb{K}_2,W)$ two vector spaces over different fields ( as an example: $\mathbb{K}_1=\mathbb{C}$ and $\mathbb{K}_2=\mathbb{R}$).
We can generalize the notion of  a linear transformation $T:\mathbf{V}\to\mathbf{W}$ for such two space?
My idea is that we have no problems for additivity:
$$
T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+ T (\mathbf{y})
$$
but we have some truble with homogeneity since for $T(\alpha \mathbf{x})$ we cannot define $\alpha T(\mathbf{x})$.
It seems that we must have some function $\lambda :\mathbb{K}_1 \to \mathbb{K}_2 $ so that we can write something as
$$
T(\alpha \mathbf{x})=\lambda(\alpha)T(\mathbf{x})
$$ 
but there is some ''natural'' definition of such function $\lambda$ that preserve the intuitive meaning of linearity?

REPLY [5 votes]: One thing you can write down is a pair consisting of a morphism $f : K_1 \to K_2$ of fields and a morphism $T : V \to W$ of abelian groups such that
$$T(ax) = f(a) T(x).$$
This is sometimes useful in the case that $K_1 = K_2$ is a Galois extension of some field $k$ and $f$ is an element of the Galois group; $T$ is then called a "semilinear" map. More generally, $K_1, K_2$ can be arbitrary rings and $V, W$ can be modules over those rings; this is part of a useful fibered category.