\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
For each of the following sets, decide whether the natural operations endow it with the indicated vector space structure. Justify briefly.
  1. \(E_1 = \mathbb{R}^3\) as an \(\mathbb{R}\)-vector space, with the usual operations.
  2. \(E_2 = \{(x, y) \in \mathbb{R}^2 : x \ge 0\}\) as an \(\mathbb{R}\)-vector space, with the usual operations of \(\mathbb{R}^2\).
  3. \(E_3 = \mathbb{C}\) as an \(\mathbb{R}\)-vector space, with the addition of \(\mathbb{C}\) and scalar multiplication \(\lambda \cdot z := \lambda z\) for \(\lambda \in \mathbb{R}\).

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