Let $S_{22}$ be the vector space of $2\times 2$ symmetric matrices. Build the matrix representation of the linear transformation $\ltdefn{T}{P_2}{S_{22}}$ relative to the bases $B$ and $C$ and then use this matrix representation to compute $\lt{T}{3+5x-2x^2}$. \begin{align*} B&=\set{1,\,1+x,\,1+x+x^2} & C&=\set{ \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix},\, \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix},\, \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} }\\ \lt{T}{a+bx+cx^2}&= \begin{bmatrix} 2a-b+c & a+3b-c \\ a+3b-c & a-c \end{bmatrix} \end{align*}