19.3.31 Linear Operators

Condition: 1) to prove that the operator \ (\ hat {a} \) is a linear operator in space \ (p_ {n} \) the degree not higher than \ (n \), 2) find the operator matrix \ (\ hat {a} \) in the canonical basis \ (p_ {n} \), whether there is a reverse operationer executive operationer. \ (\ HAT {A}^{-1} \)? If so, find his matrix, 4) Find the image, nucleus, rank and defect of the operator \ (\ hat {a} \). \ [n = 2, \ quad (\ hat {a} p) t = \ frac {d} {d t} [t p (t+1)] \]