Why is the linear transformation of the integral not onto? I know its one-to-one, but why not onto? explain?

Question

First lets clarify the space over which this is happening. Let's say its the vector space V of polynomials. T:V-->V
I'm looking for the simplest easy to understand answer.
Is it because there are no elements in the domain whose image are constants? And constants are in the codomain but not in the range.....is that right...?

Answer 1

Your T probably gives the term-by-term integral of polynomials, since - by linearity - T(0)=0 necessarily. So T can never attain a non-zero constant value.