Definition:Ordered Dual Basis

Theorem
Let $R$ be a commutative ring.

Let $\left({G, +_G, \circ}\right)_R$ be an $n$-dimensional module over $R$.

Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$.

Let $G^*$ be the algebraic dual of $G$.

Then there is an ordered basis $\left \langle {a'_n} \right \rangle$ of $G^*$ satisfying $\forall i, j \in \left[{1 \,. \, . \, n}\right]: a'_i \left({a_j}\right) = \delta_{i j}$.

This ordered basis $\left \langle {a'_n} \right \rangle$ of $G^*$ is called the ordered basis of $G^*$ dual to $\left \langle {a_n} \right \rangle$, or the ordered dual basis of $G^*$.

Proof
Since $\left\{{1_R}\right\}$ is a basis of the $R$-module $R$, by Product of Linear Transformations this basis as described exists.