Chapter 5.7, Problem 16E

Identify $\mathcal{N}\left(T\right)$ and $\mathcal{R}\left(T\right)$ for the linear transformation $T$ given in Example 3.

Example 3 Let $T:C\left[0,1\right]\to R^1$ be defined by

$T\left(f\right)={\displaystyle {\int }_0^{1}f\left(t\right)dt}$

Prove that $T$ is a linear transformation.