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# Dual vector ()

nosplit "Dual space" words: 174
Dual vectors are the members of a dual space.
In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis: $$e1​e2​e3​e1e2e3​∈V∈V∈V∈V∗∈V∗∈V∗​ (6)$$ The dual basis vectors are defined to "pick the corresponding coordinate" out of elements of V. E.g.: $$e1(4,−3,6)e2(4,−3,6)e3(4,−3,6)​=4=−3=6​ (7)$$ By expanding into the basis, we can put this more succinctly with the Kronecker delta as: $$ei(ej​)=δij​ (8)$$
Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector as: $$f=fi​ei (9)$$
In the context of quantum mechanics, the bra notation is also used for dual vectors.