Question: Let V be a 2-dimensional complex vector space with basis \(B=\left\{ \mathbf{a},\mathbf{b} \right\}\). Show that \(\left\{ \mathbf{a},i\mathbf{a},\mathbf{b},i\mathbf{b} \right\}\) is a basis for the corresponding real vector space. That is, show that for each element \(\mathbf{v}\in V\) there are unique real numbers \({{x}_{j}}\,\,\left( 1\le j\le 4 \right)\) with \(\mathbf{v}={{x}_{1}}\mathbf{a}+{{x}_{2}}i\mathbf{a}+{{x}_{3}}\mathbf{b}+{{x}_{4}}i\mathbf{b}\).

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