t(a_m)$.

Additionally, for each vertex $i$, there is a path of length $0$ starting and ending at $i$ and denoted by $e_i$. We call it either the *trivial path* or the *lazy path* at $i$. If $w$ is any path, then $w=e_{t(w)}w = we_{s(w)}$.

Let $Q$ be a quiver. A *representation of $Q$* is a tuple $V=(V_i, V_a)_{i\in Q_0, a\in Q_1}$, where

-   for each vertex $i$ in $Q$, $V_i$ is a $k$-vector space, and

-   for each arrow $a$ in $Q$, $V_a:V_{s(a)}\to V_{t(a)}$ is a $k$-linear map.

A representation $V$ is said to be *finite-dimensional* if all the vector spaces $V_i$ are finite-dimensional; in that case, the *dimension vector* of $V$ is ${\underline{\dim}\,}V = (\dim V_i)_{i\in Q_0}$. If $w=a_m\cdots a_1a_0$ is a path in $Q$, we write $V_w = V_{a_m}\circ \cdots \circ V_{a_1}\circ V_{a_0}$.

In these notes, all representations will be finite-dimensional.

Let $V$ and $W$ be two representations of a quiver $Q$. A *morphism of representations from $V$ to $W$*, denoted by $f:V\to W$, is a tuple $f=(f_i)_{i\in Q_0}$, where

-   for each vertex $i$ of $Q$, $f_i:V_i\to W_i$ is a $k$-linear map, and

-   for each arrow $a$ of $Q$, we have that $W_a\circ f_{s(a)} = f_{t(a)}\circ 