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If $V$ is a real vector space, its complexification $\Vc$ is the complex vector space consisting of elements $x + iy$ , where $x, y \in V$ . Vector addition and scalar multiplication by complex numbers are defined in the obvious way:
If
 is a basis for $V$ , then
 is a basis for $\Vc$ . Naturally, $x + i0 \in \Vc$ is often written just as $x$ .
So, for example, the complexification of $\real^n$ is (isomorphic to) $\complex^n$ .
If $T\colon V \to W$ is a linear transformation between two real vector spaces $V$ and $W$ , its complexification $\Tc\colon \Vc \to \Wc$ is defined by
It may be readily verified that $\Tc$ is complex-linear.
If
 is a basis for $V$ ,
 is a basis for $W$ , and $A$ is the matrix representation of $T$ with respect to these bases, then $A$ , regarded as a complex matrix, is also the representation of $\Tc$ with respect to the corresponding bases in $\Vc$ and $\Wc$ .
So, the complexification process is a formal, coordinate-free way of saying: take the matrix $A$ of $T$ , with its real entries, but operate on it as a complex matrix. The advantage of making this abstracted definition is that we are not required to fix a choice of coordinates and use matrix representations when otherwise there is no need to. For example, we might want to make arguments about the complex eigenvalues and eigenvectors for a transformation $T\colon V \to V$ ,
while, of course, non-real eigenvalues and eigenvectors, by definition, cannot exist for a transformation between real vector spaces. What we really mean are the eigenvalues and eigenvectors of $\Tc$ .
Also, the complexification process generalizes without change for infinite-dimensional spaces.
Finally, if $V$ is also a real inner product space, its real inner product can be extended to a complex inner product for $\Vc$ by the obvious expansion: $$ \langle x+iy, u+iv \rangle = \langle x, u \rangle + \langle y, v \rangle + i(\langle y, u \rangle - \langle x, v \rangle)\,. $$ It follows that $\norm{x+iy}^2 = \norm{x}^2 + \norm{y}^2$ .
More generally, for a real normed vector space $V$ , the equation $$ \norm{x+iy}^2 = \norm{x}^2 + \norm{y}^2 $$ can serve as a definition of the norm for $\Vc$ .
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- Vladimir I. Arnol'd. Ordinary Differential Equations. Springer-Verlag, 1992.
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