Vector space/Basis/Subbasis/Orthogonal space/Example

Let ${}V$ be a finite-dimensional ${}K$-vector space with a basis ${}v_{i}$, ${}i\in I$, and the corresponding dual basis ${}v_{i}^{*}$, ${}i\in I$. Let

${}U=\langle v_{j},\,j\in J\rangle \,$

for a subset ${}J\subseteq I$. Then

${}U^{\perp }=\langle v_{i}^{*},\,i\not \in J\rangle \,.$