Editing Complexification (section)

== Linear transformations ==
Given a real [[linear transformation]] {{math|''f'' : ''V'' → ''W''}}  between two real vector spaces there is a natural complex linear transformation
:<math>f^{\Complex} : V^{\Complex} \to W^{\Complex}</math>
given by
:<math>f^{\Complex}(v\otimes z) = f(v)\otimes z.</math>
The map <math>f^{\Complex}</math> is called the '''complexification''' of ''f''. The complexification of linear transformations satisfies the following properties
*<math>(\mathrm{id}_V)^{\Complex} = \mathrm{id}_{V^{\Complex}}</math>
*<math>(f \circ g)^{\Complex} = f^{\Complex} \circ g^{\Complex}</math>
*<math>(f+g)^{\Complex} = f^{\Complex} + g^{\Complex}</math>
*<math>(a f)^{\Complex} = a f^{\Complex} \quad \forall a \in \R</math>

In the language of [[category theory]] one says that complexification defines an ([[additive functor|additive]]) [[functor]] from the [[category of vector spaces|category of real vector spaces]] to the category of complex vector spaces.

The map {{math|''f''{{i sup|'''C'''}}}} commutes with conjugation and so maps the real subspace of ''V''{{i sup|'''C'''}} to the real subspace of {{math|''W''{{i sup|'''C'''}}}} (via the map {{math|''f''}}). Moreover, a complex linear map {{math|''g'' : ''V''{{i sup|'''C'''}} → ''W''{{i sup|'''C'''}}}} is the complexification of a real linear map if and only if it commutes with conjugation.

As an example consider a linear transformation from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} thought of as an {{math|''m''×''n''}} [[matrix (mathematics)|matrix]]. The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from {{math|'''C'''<sup>''n''</sup>}} to {{math|'''C'''<sup>''m''</sup>}}.