Basis of The Dual Space in Finite Dimensions
Let the vector space V have a basis, not necessarily orthogonal. Then the dual space V* has a basis called the dual basis defined by the special property that
Or, more succinctly,
where δ is the Kronecker delta. Here the superscripts of the basis functionals are not exponents but are instead contravariant indices.
A linear functional belonging to the dual space can be expressed as a linear combination of basis functionals, with coefficients ("components") ui,
Then, applying the functional to a basis vector ej yields
due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then
that is
This last equation shows that an individual component of a linear functional can be extracted by applying the functional to a corresponding basis vector.
Read more about this topic: Linear Functional, Bases in Finite Dimensions
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