Reflection Functor of a Quiver

I think I have figured it out;

let $h:\bigoplus_{\substack{\alpha\in Q_{1}\\ \alpha:j\to i}} X_{j} \to X_{i}$ be the third map in the sequence then we define $h(x_{1},\ldots,x_{n}):=f_{1}x_{1}+\cdots+f_{n}x_{n}$, where $f_{j}:X_{j}\to X_{i}$ is the map associated to the edge $\alpha:j\to i\in Q_{1}$. Then we have $$\ker(h)=\{(x_{1},\ldots,x_{n})\in\bigoplus_{\substack{\alpha\in Q_{1}\\ \alpha:j\to i}} X_{j}: f_{1}x_{1}+\cdots+f_{n}x_{n}=0\}.$$

Although I still don't know anything about the dimension in general.

If the representation is indecomposable, the map $\oplus_{\alpha:j \to i} f_\alpha$ is surjective (otherwise you could split off a complement to its image as a summand) and hence the dimension of its kernel is the sum over the incoming arrows of the dimensions of the vector spaces at their sources minus the dimension of the vector space at $i$.