The linear transformation $M=$$\left[ \begin{array} { c c } { \cos ( 40 ^ \circ ) } & { \sin ( 40 ^ \circ ) } \\ { -\sin ( 40 ^ \circ ) } & { \cos ( 40 ^ \circ ) } \end{array} \right]$.  

1. Algebraically describe the effect of $M$ on $(x, y)$.

   Step (a):

   $\left[ \begin{array} { c c } { \cos ( 40 ^ \circ ) } & { \sin ( 40 ^ \circ ) } \\ { -\sin ( 40 ^ \circ ) } & { \cos ( 40 ^ \circ ) } \end{array} \right] \left[ \begin{array} { c c} { x } \\{y} \end{array} \right]=$

2. Geometrically describe $M$.

   Hint (b):

   Try transforming the points $(1, 0)$ and $(0, 1)$. Where do they end up in relation to where they started?

3. Write the matrix for $M^{−1}$.

   Step (c):

   $\left[ \begin{array} { c c } { \cos ( 40 ^ \circ ) } & { \sin ( 40 ^ \circ ) } \\ { -\sin ( 40 ^ \circ ) } & { \cos ( 40 ^ \circ ) } \end{array} \right] \left[ \begin{array} { c c} { a }&{b} \\{c} &{d}\end{array} \right]= \left[ \begin{array} { c c} { 0 }&{1} \\{1} & {0}\end{array} \right]$

   Complete the multiplication and solve for $a, b, c,$ and $d$.