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Every complex vector space $(E,+,\cdot_\mathbb{C})$ can be turned into a real vector space by defining the scalar multiplication $\cdot_\mathbb{R}$ by real numbers as follows: $$a\cdot_\mathbb{R}v=(a+i0)\cdot_\mathbb{C}v,$$ for $v\in E$ and $a\in\mathbb R$.

How can I get a (real) basis of $(E,+,\cdot_{\mathbb R})$ form a (complex) basis of $(E,+,\cdot_{\mathbb C})$?

It is clear the converse. Given $\{e_1,\ldots,e_n\}$ basis of $(E,+\cdot_{\mathbb R})$, then $\{e_1,\ldots,e_n,i e_1,\ldots, i e_n\}$ must be a basis of $E$.

An idea could be take in count the canonical identification of $\mathbb C^n$ with $\mathbb R^{2n}$.

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If $\{e_1,\ldots,e_n\}$ is a basis of $(E,+,\cdot_{\mathbb C})$, then $\{e_1,\ldots,e_n,i e_1,\ldots, i e_n\}$ is a basis of $(E,+,\cdot_{\mathbb R})$.

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