$T:P_3\rightarrow \mathbb{R}$ given: $T(p)= \int_0^1x^2p(x)dx$. Prove that T is a linear transformation, and find a basis for its kernel.
DO NOT SOLVE
My textbook explain using only very abstract terms, and as I'm new to the concept; I'm having difficulty in applying the idea of 'linear transformations'.
Questions:
what is the term $P_3$... it is not defined. I'm guessing a polynomial of at most 3rd degree. Further is $p(x)$ then also a polynomial of 3rd degree?
Any hints/ advise would be much appreciated.