So I have to determine if the following is a linear transformation: $$T: F(I) \rightarrow F(I)$$ defined by: $$T(f) = 2f$$
I know that if you let $T: V\rightarrow W$ be a linear transformation. Then:
a. $T(cu) = cT(u)$ for all $v$ in $V$.
b. $T(u-v) = T(u)-T(v)$ for all $u$ and $v$ in $V$.
So I think I am supposed to let $f_1,f_2$ be in $F$ (or maybe $I$??) and let a scalar $c$ be a real number as well and put them into a, b and solve them for the proof but I'm not sure how I would apply this for this question since the definition of the transformation is $T(f) = 2f$.
Any help would be appreciated. Thanks.
Your "vectors" are elements of $F(I)$, which are functions on the interval(?) $I$. Note that the zero "vector" is actually a function that sends everything to zero.
Your scalars are probably real numbers. They need to be in a field (e.g. real numbers, complex numbers) and should not be in $F(I)$.
Finally, I think you should use the definition of linearity that Ilya posted in the comment above; in particular I don't think $T(cv)=c T(v)$ is implied by the three conditions that you wrote.