1.7.12 Linear transformations

condition: 1) Prove that \( \widehat{A}- \) is a linear operator in the space \( \mathbb{P}_{n} \) of polynomials of degree at most \( n \). 2) Find its matrix in the canonical basis. 3) Is there an inverse operator to \( \widehat{\mathrm{A}} \) ? If yes, then find its matrix in the same basis. 4) Find the kernel of the operator \( \widehat{A} ​​\), that is, the set \( \operatorname{Ker} \widehat{\mathrm{A}}=\left\{p(t) \in, \mathbb{P}_{n}:(\widehat{\mathrm{A}} p)(t) \equiv 0\right\} \). \[ n=3, \quad(\widehat{\mathrm{A}} p)(t)=t \cdot p^{\prime}(t+1) \]