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In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I , the dual set of B is a set B of vectors in the dual space V with the same index set I such that B and B form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V. If it does span V, then B is called the dual basis or reciprocal basis for the basis B.
Denoting the indexed vector sets as B = { v i } i ∈ I {\displaystyle B=\{v_{i}\}_{i\in I}} and B ∗ = { v i } i ∈ I {\displaystyle B^{*}=\{v^{i}\}_{i\in I}} , being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in V on a vector in the original space V:
where δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker delta symbol.