Which are Bases?

Algebra Level 3

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

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