Projective dimension is defined as the length of a minimal projective resolution of a module. As an example, take the $\mathbb{A}_3$ quadratic algebra:

$$\bullet \to_a \bullet \to_b \, ,$$

with the relation $ba = 0$. Label the simple modules represented by vertices from left to right by (1), (2), and (3). The projective resolutions are:

```
(1) ← P_1 ← P_2 ← P_3=(3)
(2) ← P_2 ← (3)
(3)
```

Note: Quiver path algebras with no relations (and therefore non-quadratic) are called "hereditary" and the minimal projective resolutions are always of length 1. Quadratic algebras will only ever have minimal resolutions of length 2. Infinite dimensional algebras with relations will often have infinite length projective resolutions, and some may even pe periodic.