19.3.16 Linear operators

condition: Given the operator \[ A x(t)=3 x(t)+(7 t-3) x^{\prime}(t)+\int_{0}^{t}(4 s-3) x^{\prime \prime}(s) d s \] in the space \( P^{2} \) polynomials of degree \( \leq 2 \). Check that \( A ; P^{2} \rightarrow P^{2} \) is a linear operator and find: a) the matrix of this operator in the canonical basis \( \left\{1, t, t^{2}\right\} \) b) the matrix of this operator in the basis \[ \left\{1-2 t, 3 t+t^{2}, 2+3 t^{2}\right\} \] c) the kernel \( \operatorname{Ker} A \) of the operator \( A \); d) the kernel \( \operatorname{Ker} A \) of the operator \( A: C^{2}[0,1] \rightarrow C^{2}[0,1] \).