Consider $T : C[0,1] \longrightarrow C[0,1]$ defined by $T(f(x)) = f^′(x)−f(x)$. I like this linear transformation because it's null space is functions of the form $ce^x$ for $c \in \mathbb{R}$ on the interval $[0,1]$. The range, on the other hand, I'm not quite as certain about. We are looking at all functions that can be written as a difference of a continuous function and it's derivative. Is this all of $C[0,1]$?
How can I convince myself of this?