Suppose that $V_1$ is a vector space over the field $K_1$ and $V_2$ is a vector space over the field $K_2$. What is the definition of an isomorphism between these two vector spaces?
My best guess would be that $K_1$ and $K_2$ have to be isomorphic as fields. Say $\xi:K_1\to K_2$ is an isomorphism. And furthermore, we must have a bijective function $\sigma:V_1 \to V_2$ with
$$\sigma(\lambda v_1 + \mu v_2) = \xi(\lambda)\sigma(v_1) + \xi(\mu)\sigma(v_2)$$ for all $v_1,v_2 \in V_1$ and all $\lambda, \mu \in K_1$.
Is this correct, or have I missed or added extra conditions?
Best Answer
I don't really know what it means for a definition to be correct. Certainly this is a sensible definition (e.g. it defines an "equivalence relation" on the "set" of vector spaces). It is the notion of isomorphism that one gets in the following category: objects are pairs $(k, V)$ of a field and a vector space over that field, and morphisms are pairs $(\phi, T)$ of field morphisms $\phi : k_1 \to k_2$ and maps $T : V_1 \to V_2$ such that $$T(av + bw) = \phi(a) T(v) + \phi(b) T(w).$$
However, when most people talk about isomorphism of vector spaces, they almost always work implicitly with a fixed base field $k$. This means more than that $k_1, k_2$ are isomorphic; it means that we have fixed an isomorphism between them.