Dual basis

In linear algebra, given a vector space ${\displaystyle V}$ with a basis ${\displaystyle B}$ of vectors indexed by an index set ${\displaystyle I}$ (the cardinality of ${\displaystyle I}$ is the dimension of ${\displaystyle V}$), the dual set of ${\displaystyle B}$ is a set ${\displaystyle B^{*}}$ of vectors in the dual space ${\displaystyle V^{*}}$ with the same index set ${\displaystyle I}$ such that ${\displaystyle B}$ and ${\displaystyle B^{*}}$ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span ${\displaystyle V^{*}}$. If it does span ${\displaystyle V^{*}}$, then ${\displaystyle B^{*}}$ is called the dual basis or reciprocal basis for the basis ${\displaystyle B}$.

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 ${\displaystyle V^{*}}$ on a vector in the original space ${\displaystyle V}$:

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

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