Prove that the linear transformation $S$ is invertible. Then find a formula for the inverse linear transformation, $\ltinverse{S}$, by employing a matrix inverse. \begin{equation*} \ltdefn{S}{P_1}{M_{12}},\quad \lt{S}{a+bx}= \begin{bmatrix} 3a+b & 2a+b \end{bmatrix} \end{equation*}