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Let \(E = \mathbb{R}^2\). Define a new « addition » \(\boxplus\) and a new scalar multiplication \(\boxdot\) by $$ \begin{aligned} (x, y) \boxplus (x', y') &:= (x + x', y + y' + 1), \\ \lambda \boxdot (x, y) &:= (\lambda x, \lambda y + \lambda - 1). \end{aligned} $$ Is \((E, \boxplus, \boxdot)\) an \(\mathbb{R}\)-vector space? Justify.
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