## The space of positive real numbers

Let $$\mathbb R^+$$ be all the positive real numbers and define operation $$\oplus$$ on $$\mathbb R^+$$ and $$\odot: \mathbb Q\times \mathbb R^+\to \mathbb R^+$$ as follows: $u\oplus v = uv, \hspace{2cm} a\odot u = u^a$ Is it true, that $$(\mathbb R^+,\oplus,\odot)$$ is a vector space over the rational numbers $$\mathbb Q$$?
Let $$\mathbf 0=1$$ be the neutral element and (SO) proceed as follow $$u\oplus \mathbf 0= u \oplus 1 = u\cdot 1=u$$.
For the axiom (SI) lets choose the oposite vector as $$\ominus u=\frac1u\in \mathbb R^+$$ and then we proceed with
$$u\oplus (\ominus u)=u\cdot \frac1u=1=\mathbf 0$$.