Dual basis

In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimensionality of V), its dual set 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 for the basis B.

Denoting the indexed vector sets as ${\displaystyle B=\{v_{i}\}_{i\in I}}$ and ${\displaystyle B^{*}=\{v^{i}\}_{i\in I}}$, 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:

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