Consider the linear transformation $\ltdefn{S}{M_{12}}{P_1}$ from the set of $1\times 2$ matrices to the set of polynomials of degree at most 1, defined by \begin{equation*} \lt{S}{\begin{bmatrix}a & b\end{bmatrix}}=(3a+b)+(5a+2b)x \end{equation*} Prove that $S$ is invertible. Then show that the linear transformation \begin{equation*} \ltdefn{R}{P_1}{M_{12}},\quad \lt{R}{r+sx}=\begin{bmatrix}(2r-s) & (-5r+3s)\end{bmatrix} \end{equation*} is the inverse of $S$, that is $\ltinverse{S}=R$.