Question 1167650
We have to show that the inner product defined by ⟨𝑢,𝑣⟩ = {{{(Au)^T*(Av)}}} produces the expression:

⟨𝑢,𝑣⟩ = {{{5u[1]v[1]-u[1]v[2]-u[2]v[1]+10u[2]v[2]}}}.



Let {{{u = (matrix(2,1,u[1],u[2]))}}} and {{{v = (matrix(2,1,v[1],v[2]))}}}.

These imply that {{{Au = (matrix(2,2,2,1,-1,3))*(matrix(2,1,u[1],u[2]))=(matrix(2,1,2u[1]+u[2],-u[1]+3u[2]))}}} and {{{Av = (matrix(2,2,2,1,-1,3))*(matrix(2,1,v[1],v[2]))=(matrix(2,1,2v[1]+v[2],-v[1]+3v[2]))}}}.

These give {{{(Au)^T*(Av)=(matrix(1,2,2u[1]+u[2],-u[1]+3u[2]))*(matrix(2,1,2v[1]+v[2],-v[1]+3v[2]))=( 2u[1]+u[2])*(2v[1]+v[2])+( -u[1]+3u[2])*(-v[1]+3v[2]) = 5u[1]v[1]-u[1]v[2]-u[2]v[1]+10u[2]v[2]}}}

The proof is complete.