Quivers and path algebras

The quiver $Q$ has vertices $1, 2, 3$ and arrows $\alpha : 1 \to 2$, $\beta : 2 \to 3$ and $\gamma : 1 \to 3$. What is $\dim_k kQ$?

For which of these finite quivers is the path algebra $kQ$ finite-dimensional? (Choose all that apply.)

In $kQ$ for $Q : 1 \xrightarrow{\alpha} 2 \xrightarrow{\beta} 3$, with the lesson's convention ($pq$ = first $p$, then $q$), which products are non-zero? (Choose all that apply.)

How many indecomposable representations, up to isomorphism, does the linearly oriented quiver $1 \to 2 \to \cdots \to 6$ have?

A representation $V$ of $1 \to 2 \to 3$ has dimension vector $(2, 3, 2)$, and the composite maps have ranks $r_{12} = 2$, $r_{23} = 2$, $r_{13} = 1$. What is its decomposition into interval modules?

In the proof that representations of $Q$ are the same as $kQ$-modules, where is it used that $Q$ has finitely many vertices?

Which Lean statement expresses the half of Theorem 2 proved in cycle_infinite (for a vertex a of a quiver V)?

For an acyclic quiver with vertices $0, \dots, n-1$ and a list of arrows (i, j) (from $i$ to $j$, repeats allowed), write projective_dims(n, arrows, a) returning the dimension vector of the projective representation $P(a)$: entry $b$ is the number of paths from $a$ to $b$ (including the trivial path when $b = a$).