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.