Use a matrix representation to find a basis for the kernel of the linear transformation $L$. \begin{equation*} \ltdefn{L}{M_{22}}{P_2},\quad \lt{T}{\begin{bmatrix}a&b\\c&d\end{bmatrix}}= (a+2b+4c+d)+(3a+c-2d)x+(-a+b+3c+3d)x^2 \end{equation*}