= 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 $(y-x)^{k-1}=\sum_{i=j-k+1}^jN_i(x)\psi_i(y)$. Evaluation of the dual <polynomials> at $t_j,t_{j-1},\ldots,t_{j-k+1}$ 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.
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