Linear Transformations and Isomorphisms

1. Find out which of the transformations are linear. For those that are linear, determine whether they are isomorphisms. T(f(t)) = f'(t) + t^2 from P2 to P2

2. To be linear, T(f(t)+g(t))=T(f(t)) + T(g(t)), kT(f(t))=T(f(kt))

3. After testing for linearity, I am thinking that the equation does not fulfill the requirements and therefore there are no isomorphisms, but I'm not sure if I did it right. First, I said that:
T(f(t)+g(t))=(f'(t)+g'(t))+ t^2
and T(f(t) + T(g(t))= f'(t)+t^2+g'(t)+t^2=f'(t)+2(t^2)+g'(t)
which is not equal to the first, therefore it is not linear. Am I going in the right direction with this?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

 PhysOrg.com science news on PhysOrg.com >> Hong Kong launches first electric taxis>> Morocco to harness the wind in energy hunt>> Galaxy's Ring of Fire
 Recognitions: Homework Help Science Advisor You are doing it right. It's not linear.

 Tags isomorphisms, linear alegbra, transformations