I am proving that $\mathbb R$ is a vector space over $\mathbb Q$.
So far, I have stated that vector addition and scalar multiplication trivially hold in $\mathbb R$. I then showed that $(\mathbb R, +)$ is an abelian group.
Now I am trying to show distributivity of scalar multiplication, but I feel it's trivial to the point where I can't really prove it.
This is what I have:
Let $\alpha, \beta \in \mathbb Q$ and $x,y \in \mathbb R$.
Then, $(\alpha+\beta)\cdot x = \alpha \cdot x+ \beta \cdot x$ and $\alpha \cdot(x+y)=\alpha \cdot x+ \alpha \cdot y$
by the usual operations in $\mathbb R$.
Is this correct? I feel it's inadequate, but I don't know how to break it down further.
Any feedback is appreciated.
Best Answer
You're right that it's pretty trivial. You know that $\mathbb R$ under normal addition is an abelian group, and the rest of the axioms follow from the fact that $\mathbb R$ under normal addition and multiplication is a ring, and $x \in \mathbb Q$ implies $x \in \mathbb R$. So your proof of distributivity is correct.