Why is the set $\{1,\sqrt {2}\}$ linearly independent over $\mathbb Q$ but not over $\mathbb R$?

Question

Why is the set $\{1,\sqrt {2}\}$ linearly independent over $\mathbb Q$ but linearly dependent over $\mathbb R$?

I have only come across these types of questions with vectors, how would you go about showing this?

Answer 1

That is because $\sqrt 2$ is irrational: if $1$ and $\sqrt 2$ were linearly dependent, $\sqrt 2$ would be rational.

On the other hand, $\mathbf R$ has dimension $1$ over itself, so any two real numbers are linearly dependent.