 Local linear independence of B-splines

The $k$ order-$k$ B-splines active on a simple-knot cell form a basis of its degree-$k-1$ polynomials. To see this from Marsden's identity, use the dual polynomials ${\psi }_i(y)={\prod }_{\ell =1}^{k-1}(y-t_{i+\ell })$. On $t_j<x<t_{j+1}$ the identity is $(y-x{)}^{k-1}={\sum }_{i=j-k+1}^j{N}_i(x){\psi }_i(y)$. Evaluation of the dual polynomials at $t_j,t_{j-1},\dots ,t_{j-k+1}$ gives a triangular coefficient matrix with nonzero diagonal. They are independent, and comparison of coefficients of $y$ in the identity gives independence of the active splines. Consequently a spline vanishes on a whole open cell exactly when all its active coefficients vanish.