Editing Complexification (section)

== Basic properties ==
By the nature of the tensor product, every vector {{math|''v''}} in {{math|''V''{{i sup|'''C'''}}}} can be written uniquely in the form
:<math>v = v_1\otimes 1 + v_2\otimes i</math>
where {{math|''v''<sub>1</sub>}} and {{math|''v''<sub>2</sub>}} are vectors in {{math|''V''}}. It is a common practice to drop the tensor product symbol and just write
:<math>v = v_1 + iv_2.\,</math>
Multiplication by the complex number {{math|''a'' + ''i b''}} is then given by the usual rule
:<math>(a+ib)(v_1 + iv_2) = (av_1 - bv_2) + i(bv_1 + av_2).\,</math>
We can then regard {{math|''V''{{i sup|'''C'''}}}} as the [[direct sum of vector spaces|direct sum]] of two copies of {{math|''V''}}:
:<math>V^{\Complex} \cong V \oplus i V</math>
with the above rule for multiplication by complex numbers.

There is a natural embedding of {{math|''V''}} into {{math|''V''{{i sup|'''C'''}}}} given by
:<math>v\mapsto v\otimes 1.</math>
The vector space {{math|''V''}} may then be regarded as a ''real'' [[linear subspace|subspace]] of {{math|''V''{{i sup|'''C'''}}}}. If {{math|''V''}} has a [[basis (linear algebra)|basis]] {{math|{{mset| ''e''<sub>''i''</sub> }}}} (over the field {{math|'''R'''}}) then a corresponding basis for {{math|''V''{{i sup|'''C'''}}}} is given by {{math|{ ''e''<sub>''i''</sub> ⊗ 1 } }} over the field {{math|'''C'''}}. The complex [[dimension (linear algebra)|dimension]] of {{math|''V''{{i sup|'''C'''}}}} is therefore equal to the real dimension of {{math|''V''}}:

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

Alternatively, rather than using tensor products, one can use this direct sum as the ''definition'' of the complexification:
:<math>V^{\Complex} := V \oplus V,</math>
where <math>V^{\Complex}</math> is given a [[linear complex structure]] by the operator {{math|''J''}} defined as <math>J(v,w) := (-w,v),</math> where {{math|''J''}} encodes the operation of “multiplication by {{mvar|i}}”. In matrix form, {{math|''J''}} is given by:
:<math>J = \begin{bmatrix}0 & -I_V \\ I_V & 0\end{bmatrix}.</math>
This yields the identical space – a real vector space with linear complex structure is identical data to a complex vector space – though it constructs the space differently. Accordingly, <math>V^{\Complex}</math> can be written as <math>V \oplus JV</math> or <math>V \oplus i V,</math> identifying {{math|''V''}} with the first direct summand. This approach is more concrete, and has the advantage of avoiding the use of the technically involved tensor product, but is ad hoc.