Editing Complexification (section)

== Formal definition ==
Let <math>V</math> be a real vector space. The '''{{em|{{visible anchor|complexification}}}}''' of {{math|''V''}} is defined by taking the [[tensor product]] of <math>V</math> with the complex numbers (thought of as a 2-dimensional vector space over the reals):

:<math>V^{\Complex} = V\otimes_{\R} \Complex\,.</math>

The subscript, <math>\R</math>, on the tensor product indicates that the tensor product is taken over the real numbers (since <math>V</math> is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, <math>V^{\Complex}</math> is only a real vector space. However, we can make <math>V^{\Complex}</math> into a complex vector space by defining complex multiplication as follows:

:<math>\alpha(v \otimes \beta) = v\otimes(\alpha\beta)\qquad\mbox{ for all } v\in V \mbox{ and }\alpha,\beta \in \Complex.</math>

More generally, complexification is an example of [[extension of scalars]] – here extending scalars from the real numbers to the complex numbers – which can be done for any [[field extension]], or indeed for any morphism of rings.

Formally, complexification is a [[functor]] {{math|Vect<sub>'''R'''</sub> → Vect<sub>'''C'''</sub>}}, from the category of real vector spaces to the category of complex vector spaces. This is the [[adjoint functor]] – specifically the [[left adjoint]] – to the [[forgetful functor]] {{math|Vect<sub>'''C'''</sub> → Vect<sub>'''R'''</sub>}} forgetting the complex structure.

This forgetting of the complex structure of a complex vector space <math>V</math> is called '''{{em|{{visible anchor|decomplexification}}}}''' (or sometimes "'''{{em|{{visible anchor|realification}}}}'''"). The decomplexification of a complex vector space <math>V</math> with basis <math>e_{\mu}</math> removes the possibility of complex multiplication of scalars, thus yielding a real vector space <math>W_{\R}</math> of twice the dimension with a basis <math>\{e_{\mu}, ie_{\mu}\}.</math><ref>{{cite book|last1=Kostrikin|first1=Alexei I.|last2=Manin|first2=Yu I.|title=Linear Algebra and Geometry|date=July 14, 1989|publisher=CRC Press|isbn=978-2881246838|page=75}}</ref>