I came across a question regarding Linear transformation stating as follows:
Let $V$ denotes the space of all real valued functions whose first derivative is continuous. And $T$ is the linear operator over V defined as follows: $$T(f) = f'(x) + \int_{0}^{x} f(t) dt$$
I couldn't find the $Ker(T)$ and also $f \in V$ the satisfies $T(f) = f$.
My thought on the first place where as follows:
The only solution for the $null$ $space$ is $0$ since we want $f'(x) + F(x) = 0$, $F(x)$ being the antiderivative, Therefore $F''(x) + f(x) = 0$ and we aren't sure regarding $F''(x)$ exists unless it's $0$
And for $T(f) = f$, I guess the only possible solution would rather be only $f = 0$ but don't know how to construct a solution for that.