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Let $\mathbf{V}(\mathbb{K}_1,V)$ and $\mathbf{W}(\mathbb{K}_2,W)$ be two vector spaces over different fields (as an example, $\mathbb{K}_1=\mathbb{C}$ and $\mathbb{K}_2=\mathbb{R}$).

Can we generalize the notion of a linear transformation $T:\mathbf{V}\to\mathbf{W}$ for such two spaces?

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 trouble 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?

J. W. Tanner
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Emilio Novati
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1 Answers1

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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.

Qiaochu Yuan
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