Let $V$ be a finite dimensional complex vector space.
I recently asked how to find a
and got some very nice answers. In particular, I was told that the map
$$c\otimes v \mapsto (\Re(c)v,\Im(c)v)$$
is a complex linear isomorphism.
Here is the problem I am having with this:
As I understand it, the scalar multiplication defined on the extension of scalars is given by
$$c'(c\otimes v) = (c'c)\otimes v$$
and with this multiplication the above map is not complex linear. However, if I use $$c'(c\otimes v)=c\otimes (c'v)$$ as my scalar multiplication then the given map is indeed complex linear.
So how do I reconcile this with the definition of scalar multiplication in an extension of scalars?