Prove that the given rule is a linear transformation, or show why it isn't.

T : C^∞(R)→R, T(f)= b∫a (f)(sin(x))dx

a = 1(bottom), b = 3 (top)

Question

Prove that the given rule is a linear transformation, or show why it isn't. 

T : C^∞(R)→R, T(f)= b∫a (f)(sin(x))dx

a = 1(bottom), b = 3 (top)

anonymous · Answer

$$T(f)=\int_a^bf(x)\sin(x)dx$$ need to check that 
$$T(f+g)=T(f)+T(g)$$ i.e. 
$$\int_a^b(f(x)+g(x))\sin(x)dx=\int_a^bf(x)\sin(x)dx+\int_a^bg(x)\sin(x)dx$$
which i believe it is

anonymous · Answer

also that 
$$T(cf)=cT(f)$$ i.e. 
$$\int_a^bcf(x)\sin(x)dx=c\int_a^bf(x)\sin(x)dx$$ again pretty clearly yes

anonymous · Answer

oh are your functions integrable?

anonymous · Answer

nvm

anonymous · Answer

ahah, so wait ... none of that applies then?