Trace in Terms of Dual Basis

Theorem
Let $R$ be a ring with unity.

Let $M$ be a free $R$-module of dimension $n$.

Let $(e_1,\ldots, e_n)$ be a basis of $M$.

Let $(e_1^*,\ldots, e_n^*)$ be its dual basis

Let $f:M\to M$ be a linear operator.

Then its trace equals:
 * $\operatorname{tr}(f) = \displaystyle \sum_{i\mathop =1}^n e_i^*(f(e_i))$

Also see

 * Trace in Terms of Orthonormal Basis