(Solved): (20 p) Consider a linear transformation \( T: P_{3}(\Re) \rightarrow P_{3}(\Re) \) defined by \[ T ...

(20 p) Consider a linear transformation \( T: P_{3}(\Re) \rightarrow P_{3}(\Re) \) defined by \[ T(f(x))=f^{\prime}(x)+f^{\prime \prime}(x) \] Test \( T \) for diagonalizability. If \( T \) is diagonalizable, find the ordered basis \( \beta \) such that \( [T]_{\beta} \) is a diagonal matrix.