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I've just learned a bit about the complexification of a real vector space V to include scalar multiplication by complex numbers. A bit of confusion has ensued, which I am hoping someone can help me with conceptually: 1) how does one generate a basis for the new space Vc? It seems that one obtains the basis by somehow extending the basis for V, but I am very confused about this. In fact, I'm not exactly sure how vectors in the new space should be defined at all. :yuck:

2) does anyone know how one would prove that the dim(Vc)=dim(V)? I'm not asking for homework; I've just heard that this is the case, but haven't seen anything proved.

3) Under what circumstances would one want to complexify V? anyone have some good examples?

Thanks.

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# Complexification of a vector space

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