Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 5 / 2 / b / Solution

Substitute the B-spline expansion $s={\sum }_{j=1}^n{a}_j{M}_j$ into $(M_i,s)=k!{\gamma }_i$. The coefficients solvewith the Gram matrix and right-hand sideThe linearly independent B-splines have a positive-definite Gram matrix, so this system has a unique coefficient vector.