Your answers are essentially correct but your notation is wrong. What you should have written, for example for (a), is
.
It is not correct to write . Do you claim that ? That makes no sense.
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