Past exam of the mathematical methods I course of the University of Oxford / 2018 / 10 / c / Solution

The two reciprocal bases satisfy $⟨u_i,w_j⟩={\delta }_{ij}$. Expanding each vector in its specified basis and using real bilinearity givesThe simple sum-of-products formula holds because the two coordinate vectors use paired reciprocal bases. It does not require either basis to be orthogonal, and would generally fail if both sets of components were taken in the same nonorthogonal basis.