This is an exercise of Advanced linear algebra of Roman
Find an abelian group $V$ and a field $F$ such that $(V,F)$ can define two distinct vector spaces.
That is: I must find two distinct definitions of scalar multiplication. What I did was choose $V:=(\Bbb R^2,+)$ as the abelian group, and $F:=\Bbb C$ as the field, and defined scalar multiplication by
$(a+bi)\cdot (x,y)=(ax-by,by+ax)$, this defines a vector space because its just the complexification of $\Bbb R$ as a vector space.
$(a+bi)\cdot (x,y)=(by+ax,ax-by)$ this is symmetric from the above definition.
But I dont found other example. Can someone show me a non so trivial example, or provide me a general rule to define such alternative scalar multiplications?
UPDATE: it seems that, from a known scalar multiplication defined by $d$, I can define an alternative scalar multiplication by $d\circ f$, where $f:V\to V$ is a group automorphism.
