Which are Bases?

Algebra

Which of the following are "bases" in the way described?

$(2,\,3)$ and $(-1,\,1)$ generate $\mathbb{R}^2$ over the real numbers.
$\dots,\,2^{-2},\,2^{-1},\,2^0,\,2^1,\,2^2,\,\dots$ generate real numbers by using their binary representation over $\mathbb{F}_2$ .
$1$ and $\sqrt{2}$ generate the algebraic numbers over the rationals.