I was having linear algebra class and we have been discussing about a possible group homomorphism that might allow mapping between two vector spaces over two different fields
This is also an extension of this question
Suppose we have vector spaces $V$ and $W$ over some general field $\mathbb{F}_1$ and $\mathbb{F}_2$ and $T$ is a (linear) map from $V$ to $W$
In order to get around the issue of this vector space axiom becoming undefined because of c being in different fields
$$T(c\mathbf{x})=cT(\mathbf{x})$$
What's the issue in doing this (adapting the definition of group homomorphism, where there are two groups $(G,@)$ and $(H,*)$)?
$$\phi (a @b)=\phi(a)*\phi(b)$$
to the context of vector space (where the fields are defined as $(\mathbb{F}_1,+,*)$ and $(\mathbb{F}_2,",@)$)
$$T(c_\mathbb{F_1}*\mathbf{x})=T(c_\mathbb{F_1})@T(\mathbf{x})=c_\mathbb{F_2}@T(\mathbf{x})$$
(The two cs are different because they are elements of different fields)
It seems valid as long every element in $\mathbb{F}_2$ can be mapped from at least one in
$\mathbb{F}_1$. What subtleties have we overlooked?
If this is valid is this still a linear algebra?
Best Answer
A map between two vector spaces over different fields cannot be linear (see this question), but can be semilinear . In this case there exists an homomorphism between the two fields $ \phi:\mathbb{F}_1 \to \mathbb{F}_2$ that is also an homomorphism between the multiplicative groups of the fields.