Dual Basis

Let \((E_i)\) be a basis of a finite dimensional vector space \(V\). Then the dual basis of the dual space \(V^*\) is determined by

\begin{equation*} \epsilon^i(E_j)=\delta^i_j. \end{equation*}

An element \(\omega\in V^*\) can then be written as

\begin{equation*} \omega=\omega_i\epsilon^i, \end{equation*}

where the components are determined by \(\omega_i=\omega(E_i)\).