Finite-dimensional vector space/Dual basis/Definition

Dual basis

Let ${}V$ denote a finite-dimensional ${}K$-vector space, endowed with a basis ${}v_{1},\ldots ,v_{n}$. Then the linear forms

${}v_{1}^{*},\ldots ,v_{n}^{*}\in {V}^{*}\,,$

defined by

${}v_{i}^{*}(v_{j})=\delta _{ij}={\begin{cases}1,{\text{ falls }}i=j\,,\\0,{\text{ falls }}i\neq j\,,\end{cases}}\,$

are called the dual basis of the given basis.