Let $S_{22}$ be the vector space of $2\times 2$ symmetric matrices. Find a basis $C$ for $S_{22}$ that yields a diagonal matrix representation of the linear transformation $R$. \begin{align*} \ltdefn{R}{S_{22}}{S_{22}},\quad \lt{R}{\begin{bmatrix}a&b\\b&c\end{bmatrix}}= \begin{bmatrix} -5a + 2b - 3c & -12a + 5b - 6c\\ -12a + 5b - 6c & 6a - 2b + 4c \end{bmatrix} \end{align*}