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), 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:





v

i


⋅

v

j


=

δ

j


i


=


{



1



if

i
=
j




0



if

i
≠
j

,









{\displaystyle v^{i}\cdot v_{j}=\delta _{j}^{i}={\begin{cases}1&{\text{if }}i=j\\0&{\text{if }}i\neq j{\text{,}}\end{cases}}}
where




δ

j


i




{\displaystyle \delta _{j}^{i}}
is the Kronecker delta symbol.

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