\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
On the set \(E = \mathbb{R}_{> 0}\) of strictly positive reals, define an addition \(\oplus\) and a scalar multiplication \(\odot\) (with scalars in \(\mathbb{R}\)) by $$ x \oplus y := xy, \qquad \lambda \odot x := x^\lambda, $$ where \(x^\lambda := e^{\lambda \ln x}\) for \(x > 0\), \(\lambda \in \mathbb{R}\) (this is the definition of arbitrary real exponentiation; it justifies the exponent laws used below). Show that \((E, \oplus, \odot)\) is an \(\mathbb{R}\)-vector space and identify the zero vector.
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