Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2011 / 1 / b / i / Solution

Let ${\epsilon }_i$ denote the coordinate linear functional dual to the $i$th vector of the standard basis. Writing a vector in the supplied basis givesSuccessive differences recover the coefficients. Hence the dual basis isThese formulas directly give $v_i^{\prime }(v_j)={\delta }_{ij}$ and exemplify the dual basis of cumulative coordinate sums.