= Solution
{c}
Let $\varepsilon_i$ denote the coordinate <linear functional> dual to the $i$th vector of the <standard basis>. Writing a vector in the supplied <basis> gives
$$
x=c_1v_1+c_2v_2+c_3v_3+c_4v_4
=(c_1+c_2+c_3+c_4,c_2+c_3+c_4,c_3+c_4,c_4)^T.
$$
Successive differences recover the coefficients. Hence the <dual basis> is
$$
\boxed{v_1'=\varepsilon_1-\varepsilon_2,\quad
v_2'=\varepsilon_2-\varepsilon_3,\quad
v_3'=\varepsilon_3-\varepsilon_4,\quad v_4'=\varepsilon_4.}
$$
These formulas directly give $v_i'(v_j)=\delta_{ij}$ and exemplify the <dual basis of cumulative coordinate sums>.
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