19.3.16 Linear Operators

Condition: DAN Operator \ [A X (T) = 3 X (T)+(7 T-3) X^{\ PRIME} (t)+\ int_ {0}^{T} (4 S-3) X^ON \ PRIME} (S) d S \ (P^{2} E)) Polynums of degree \ (\ leq 2 \). Check that \ (a; p^{2} \ rightarrow p^{2} \) linear operator and find: a) the matrix of this operator in the canonical basis \ (\ left {1, t^{2} \ right \} \) the matrix of this operator in the basis \ [[ \ left \ {1-2 t, 3 t+t^{2}, 2+3 t^{2} \ right \} \] c) core \ (\ OperatorName {Ker} A \) operator \ (A \); d) the core \ (\ Operatorname {ker} a \) operator \ (a: c^{2} [0.1] \ rightarrow c^{2} [0,1] \).