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.

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