Hi, would someone be able to check my proofs for me and tell me if they are right and if not what is wrong please?

So for the first one I said let u=p(x) and v=b(x)

T(u+v)=p(x)+b(x)=p(5)x^2+b(5)x^2=Tu+Tv

and T(ku)=(kp)(x)=kp(5)x^2=kTu

So it is a linear transformation.

For the second I said T(u+v)=p(x)+b(x)=x^2p(1/x)+x^2b(1/x)=Tp(x)+Tb(x)

and T(ku)=x^2kp(1/x)=k(x^2p(1/x))=kTp(x)

So it is also a linear transformation.

For the third I said T(kp)(x)=xkp'(x)kp''(x)=k^2xp'(x)p''(x) which does not equal KTp(x)

So it is not a linear transformation

Did I do these right?

Thanks for any help