Where is it you are stuck on this problem?

I believe proving that B is a basis is the hard bit;

Note that B forms a spanning set for your space by it's very definition. Then, (this is quite a nice proof to do) and so your basis intersects trivially with the rationals. This gives you the fact that your spanning set is linearly independent (why?).

To prove that it is a linear space, you can either hit it with a stick (go through the axioms in a boring way), or you should note that the real numbers form a vector space with the rationals as the field. Here you are just taking a subset of the reals as your basis, and so everything is nice.