Editing Dual basis (section)

{{Short description|Linear algebra concept}}
In [[linear algebra]], given a [[vector space]] <math>V</math> with a [[Basis (linear algebra)|basis]] <math>B</math> of [[Vector (mathematics and physics)|vectors]] indexed by an [[index set]] <math>I</math> (the [[cardinality]] of <math>I</math> is the [[dimension (vector space)|dimension]] of <math>V</math>), the '''dual set''' of <math>B</math> is a set <math>B^*</math> of vectors in the [[dual space]] <math>V^*</math> with the same index set <math>I</math> such that <math>B</math> and <math>B^*</math> form a [[biorthogonal system]]. The dual set is always [[linearly independent]] but does not necessarily [[Linear span|span]] <math>V^*</math>. If it does span <math>V^*</math>, then <math>B^*</math> is called the '''dual basis''' or '''reciprocal basis''' for the basis <math>B</math>.

Denoting the indexed vector sets as <math>B = \{v_i\}_{i\in I}</math> and <math>B^{*} = \{v^i\}_{i \in I}</math>, 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 <math>V^*</math> on a vector in the original space <math>V</math>:
:<math>
v^i\cdot v_j = \delta^i_j =
\begin{cases}
  1 & \text{if } i = j\\
  0 & \text{if } i \ne j\text{,}
\end{cases}
</math>
where <math>\delta^i_j</math> is the [[Kronecker delta]] symbol.