Change of Basis
6.47. Find the change-of-basis matrix \( P \) from the usual basis \( E \) of \( R^{2} \) to a basis \( S \), the change-of-basis matrix \( Q \) from \( S \) back to \( E \), and the coordinates of \( v=(a, b) \) relative to \( S \), for the following bases \( S \) :
(a) \( S=\{(1,2),(3,5)\} \).
(c) \( S=\{(2,5),(3,7)\} \).
(b) \( S=\{(1,-3),(3,-8)\} \).
(d) \( S=\{(2,3),(4,5)\} \).