Editing Four-vector (section)

=====Dual vectors=====

Applying the Minkowski tensor is often expressed as the effect of the [[dual space#Ailinear products and dual spaces|dual vector]] of one vector on the other:

<math display="block">\mathbf{A \cdot B} = A^*(\mathbf{B}) = A{_\nu}B^{\nu}. </math>

Here the ''A<sub>ν</sub>''s are the components of the dual vector '''A'''* of '''A''' in the [[dual basis]] and called the [[Covariance and contravariance of vectors|covariant]] coordinates of '''A''', while the original ''A<sup>ν</sup>'' components are called the [[Covariance and contravariance of vectors|contravariant]] coordinates.