I'm in a matrix theory class, and today we started talking about linear transformations. My professor noted that the range and domain of a linear transformation must be vector spaces over the same field. This made sense to me at first because of the fields are different, then something like $T(cx)=cT(x)$ doesn't make any sense when c is only in one of the vector spaces.
But then I thought that maybe some fields might be compatible, like for example, $\mathbb{Z}_5$ and $\mathbb{Z}_7$
Are there any linear transformations for two spaces over different fields? or Are there functions that have all the same properties as linear transformations except that the spaces involved use different fields?
If the answer is no, can you prove it?
Best Answer
If you have an injective field endomorphism $\varphi$ from field $Fa$ to field $Fb$ - said differently:
$$\forall (x, y) \in Fa \times Fa, \varphi(a+b) = \varphi(a) + \varphi(b) \space \mathrm{and} \space \varphi(a.b) = \varphi(a).\varphi(b)$$
then $\varphi(Fa)$ is a sub field of $Fb$ isomorphic to $Fa$
If $Vb$ is a vector space over $Fb$ of finite dimension, you can exhibit a base from $Vb$. You can then build a sub set (not a sub vector space) of $Vb$ from the base and the field $\varphi(Fa)$. This sub set (let us call it $Wb$) is a vector space over $\varphi(Fa)$ (*).
By identifying $Fa$ and $\varphi(Fa)$, you can indeed build a linear transformations (say $\Lambda$) from a vector space over $Fa$ (say $Va$) and $Wb$.
But you cannot call it a linear transformations from $Va$ to $Vb$ because $\Lambda(Va)$ is not a sub vector space of $Vb$ (except for the trivial $\Lambda(Va) = \{ 0_{Vb} \}$ see below). It is only a sub vector space of $Wb$.
Well to be honest, there is one trivial case where you could exhibit a pseudo linear transformation for $Va$ to $Vb$:
$$\begin{align}\Lambda: &Va \rightarrow Vb \\ &x \mapsto 0_{Vb}\end{align}$$
It respects all the properties for a linear transformation, and $\Lambda(Va) = \{ 0_{Vb} \}$ is indeed a sub vector space but it is not really interesting, and on a strict point of view, as the fields are different, it is not a true linear transformation.
(*) I suppose it could be extended to vector space of infinite dimensions, but I do not know how to prove it