= Solution
Fix $\xi_i\in[t_i,t_{i+k}]$ and choose a knot interval $I=(t_\ell,t_{\ell+1})$ adjacent to it on which $N_i$ is active. Exactly $k$ B-splines are nonzero on $I$, and their polynomial restrictions form a basis of $\mathcal P_{k-1}$. Applying the expansion from part (a) to the polynomial piece $N_j|_I$ gives
$$
N_j(t)=\sum_r\lambda_r(N_j,\xi_i)N_r(t),
\qquad t\in I.
$$
Uniqueness of coordinates in this local basis forces
$$
\lambda_i(N_j,\xi_i)=\delta_{ij}
$$
when $N_j$ is active. If $N_j$ vanishes on $I$, all of its local polynomial derivatives vanish and the same equality holds with value zero. At a knot, use either adjacent polynomial piece; the $x$-independence proved in part (a) gives the same coefficient. Hence
$$
\boxed{\lambda_i(N_j,\xi_i)=\delta_{ij},
\qquad \xi_i\in[t_i,t_{i+k}]},
$$
so the $\lambda_i$ form the <dual basis> to the B-spline basis.
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