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In mathematics, the complexification of a real vector space ''V'' is a vector space ''V''C over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for ''V'' over the real numbers serves as a basis for ''V''C over the complex numbers. == Formal definition == Let ''V'' be a real vector space. The complexification of ''V'' is defined by taking the tensor product of ''V'' with the complex numbers (thought of as a two-dimensional vector space over the reals): : The subscript R on the tensor product indicates that the tensor product is taken over the real numbers (since ''V'' is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, ''V''C is only a real vector space. However, we can make ''V''C into a complex vector space by defining complex multiplication as follows: : 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 VectR → VectC, 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 VectC → VectR from forgetting the complex structure. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complexification」の詳細全文を読む スポンサード リンク
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