Vector space/Basis/Subbasis/Orthogonal space/Example

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

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

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

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