Find matrix of $T$ if $T:P_2(x) \to P_3(x)$ is given.

Question

If $T:P_2(x) \to P_3(x)$ is such that $T(f(x))=f(x)+5\int_0^x f(t)dt$, then choosing $\{1,1+x,1-x^2\}$ and $\{1,x,x^2,x^3\}$ as bases of $P_2(x)$ and $P_3(x)$ respectively, find the matrix of T.

Answer 1

See my comment above. There is a result that I would use here. It tells me that I need to find the image of each of the basis vectors under $T$ and express these images as coordinate vectors with respect to the given basis for the image space. These vectors will be the column vectors or the required matrix.