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)

Answer 1

\[T(f)=\int_a^bf(x)\sin(x)dx\] 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

Answer 2

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

Answer 3

oh are your functions integrable?

Answer 4

nvm

Answer 5

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