19.3.31 Linear operators

condition: 1) Prove that the operator \( \hat{A} ​​\) is a linear operator in the space \( P_{n} \) of polynomials of degree at most \( n \), 2) Find the matrix of the operator \( \hat{A} ​​\) in the canonical basis \( P_{n} \), 3) Does the inverse operator exist \( \hat{A}^{-1} \) ? If yes, find its matrix, 4) Find the image, kernel, rank and defect of the operator \( \hat{A} ​​\). \[ n=2, \quad(\hat{A} ​​P) t=\frac{d}{d t}[t p(t+1)] \]