Consider an arbitrary vector space $(V, \oplus, \odot)$ over a field $F$ with
- vector addition $\oplus : V \times V \to V$ and
- scalar multiplication $\odot : F \times V \to V$,
both satisfying all the axioms defining a vector space.
Let us fix the field $F$ and the set $V$. It is obvious that the vector addition does not have to be unique. Any binary operation $\oplus$ that makes $(V, \oplus)$ an Abelian group, would actually work. But I am not so sure if this is also true for (the) scalar multiplication.
Let us fix the field $F$ and the Abelian group $(V, \oplus)$. Is the action of $F$ on $(V, \oplus)$, i.e., the scalar multiplication $\odot$ satisfying the axioms of a vector space, a unique operation?