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Is it possible to state a vector of complex numbers as a linear combination of a real numbers vector (Re(z)) and a another real numbers vector (Im(z)) multiplied by i?

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Yes, for example, $$(83+42i,96-15i)=(83,96)+i(42,-15)$$

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Every element in $\mathbb{R}^n $ can be written as a linear combination of the basis vectors $(e_i)_{i=1}^n$, where $e_i=(0.\ldots,0,1,0,\ldots,0)$ where $1$ is at the $i$th position. Now for any $z\in\mathbb{C}^n$ we have $z=x+iy$ where $x,y\in\mathbb{R}^n$ and each of the $x$ and $y$ can be expressed as a linear combination of $e_i$'s.

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