Rational Numbers form Vector Space

Theorem

Let $\Q$ be the set of rational numbers.

Then the $\Q$-module $\Q^n$ is a vector space.

It follows directly, by setting $n=1$, that the $\Q$-module $\Q$ itself can also be regarded as a vector space.

Proof

From the definition, a vector space is a unitary module whose scalar ring is a division ring.

As the $\Q$ is a field, $\Q$ is a division ring.

So the $\Q$-module $\Q^n$ fits the description.

$\blacksquare$