Dual basis of cumulative coordinate sums ($v_j=e_1+\cdots +e_j⟹v_i^{\prime }=e_i^{\prime }-e_{i+1}^{\prime }(i<n),v_n^{\prime }=e_n^{\prime }$)

For the basis $v_j=e_1+\cdots +e_j$ obtained from a basis $e_1,\dots ,e_n$, the dual basis is obtained by successive differences of the original dual basis. Indeed, if $x={\sum }_j{c}_j{v}_j$, its original coordinates are $x_i={\sum }_{j\ge i}{c}_j$. Hence $c_i=x_i-x_{i+1}$ for $i<n$, and $c_n=x_n$. These are precisely the new coordinate linear functionals. This gives an elementary example of the inverse relation between change of basis and covector transformation under a change of basis.