This is problem 2.23 from Etingof's Introduction to representation theory.
Let Q be a quiver without oriented cycles, and $P_Q$ the path algebra of $Q$. Find irreducible representations of $P_Q$ and compute $Ext^1$ between them. Classify 2-dimensional representations of $P_Q$.
I can see that irreducible representations of a quiver $Q$ without oriented cycle are exactly the 1-dimensionals. And as suggested by the author, we may compute $Ext^1(V, W)=Z^1(V, W)/B^1(V,W)$ where $V,W$ are 1-dim representations of $P_Q$. Denote $p_i$ the path whose source and target are vertex $v_i$, and $a_h$ the path corresponds to the edge $h$. Then $P_Q$ is generated by $p_i$'s and $a_h$'s.
My question is:
Since there are no oriented cycles, is it true that every path $p$ in $P_Q$ acts on an irreducible V by $0$ except for the $p_k$ for which the vector space at $v_k$ is the only non-zero vector space?