= Dual basis of cumulative coordinate sums
{title2=$v_j=e_1+\cdots+e_j\implies v_i'=e_i'-e_{i+1}'\ (i<n),\quad v_n'=e_n'$}
For the <basis> $v_j=e_1+\cdots+e_j$ obtained from a <basis> $e_1,\ldots,e_n$, the <dual basis> is obtained by successive differences of the original <dual basis>. Indeed, if $x=\sum_jc_jv_j$, its original <coordinates> are $x_i=\sum_{j\geq i}c_j$. 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>.
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