1

\begin{aligned} x (t) & = x_{0} + x_{1} t + x_{2} t^{2} \\ y (t) & = y_{0} + y_{1} t + y_{2} t^{2} \\ x (0) y (1) + x (1) y (0) & = ((x)_{ℰ})^{T} 𝔸 (y)_{ℰ} \\ x_{0} (y_{0} + y_{1} + y_{2}) + (x_{0} + x_{1} + x_{2}) y_{0} & = (\begin{array}{ccc} x_{0} & x_{1} & x_{2} \end{array}) 𝔸 (\begin{array}{c} y_{0} \\ y_{1} \\ y_{2} \end{array}) \\ 2 x_{0} y_{0} + x_{0} y_{1} + x_{0} y_{2} + x_{1} y_{0} + x_{2} y_{0} & = x_{0} (a_{11} y_{0} + a_{12} y_{1} + a_{13} y_{2}) \\ + x_{1} (a_{21} y_{0} + a_{22} y_{1} + a_{23} y_{2}) \\ + x_{2} (a_{31} y_{0} + a_{32} y_{1} + a_{33} y_{2}) \\ 𝔸 & = (\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{array}) \end{aligned}