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.