Determine if the linear transformation $\ltdefn{T}{P_2}{M_{22}}$ is (a) injective, (b) surjective, (c) invertible. \begin{equation*} \lt{T}{a+bx+cx^2}= \begin{bmatrix} a+2b-2c & 2a+2b \\ -a+b-4c & 3a+2b+2c \end{bmatrix} \end{equation*}