Let $S_{22}$ be the set of $2\times 2$ symmetric matrices. Verify that the linear transformation $R$ is invertible and find $\ltinverse{R}$. \begin{equation*} \ltdefn{R}{S_{22}}{P_2},\quad\lt{R}{\begin{bmatrix}a&b\\b&c\end{bmatrix}}= (a-b)+(2a-3b-2c)x+(a-b+c)x^2 \end{equation*}