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 VC is only a real vector space. However, we can make VC 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.
Read more about this topic: Complexification
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