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

Take the inner product with $u_i$ in $a={\sum }_j{a}_j^{\prime }{w}_j$. The reciprocal basis condition gives $a_i^{\prime }=⟨u_i,a⟩$. Expanding instead in the $u$ basis proves the reciprocal-basis coordinate transformationThus $G$ is the Gram matrix of the original basis. It is a positive-definite matrix, so this change of basis is invertible.

In the specified two-dimensional Cartesian frame,Solving the four dot product conditions givesThey are perpendicular to $u_2$ and $u_1$, respectively, with lengths adjusted so the other dot product is one. Equating the Cartesian components of $a=a_1{u}_1+a_2{u}_2=a_1^{\prime }{w}_1+a_2^{\prime }{w}_2$ givesConsequentlywhich agrees entry by entry with the Gram matrix computed directly from the two unit basis vectors.