Define the function $T: P_{2} ightarrow P_{2}$ via $T(g(x))=\left(2 X^{2}-1 ight) g^{\prime} (x)+g(x)$. a. Give $T\left(x^{2}-3 X+2 ight)$. b. Find (if possible) all functions $f(x) \in P_{2}$ for which $T(f(x)) =-2 x^{2}- X-1$. Find the eigenvalues and eigenspaces associated with the matrix $\left(\begin{array}{cc}1 & -2 \\ 2 & 1\end{array} ight) $. CS. JG. 089