Prove that the set $S$ below is a basis for the vector space of $2\times 2$ matrices, $M_{22}$. Do this by choosing a natural basis for $M_{22}$ and coordinatizing the elements of $S$ with respect to this basis. Examine the resulting set of column vectors from $\complex{4}$ and apply the Coordinatization Principle. \begin{equation*} S=\set{ \begin{bmatrix} 33 &99\\ 78 & -9 \end{bmatrix},\, \begin{bmatrix} -16 & -47\\ -36 & 2 \end{bmatrix},\, \begin{bmatrix} 10 & 27\\ 17 & 3 \end{bmatrix},\, \begin{bmatrix} -2 & -7\\ -6 & 4 \end{bmatrix} } \end{equation*}