1.7.12 Linear Transformations

Condition: 1) prove that \ (\ widehat {a}- \) a linear operator in space \ (\ mathbb {p} _ {n} \) of polynomials no higher \ (n \). 2) Find his matrix in the canonical basis. 3) Is there a reverse operator to \ (\ widehat {\ mathrm {a}} \)? If so, then find his matrix in the same basis. 4) Find the nucleus of the operator \ (\ widehat {a} \), that is, many \ (\ operatorname {ker} \ widehat {\ mathrm} = \ left \ {p (t) \ in, \ Mathbb {p} _ {n}: (\ widehat {\ mathrm {a}} p) (t) \ equiv 0 \ right \} \). \ [n = 3, \ quad (\ widehat {\ mathrm {a}} p) (t) = t \ cdot p^{\ prime} (t+1) \] \]