or $v \in E^0$, let $\delta_v$ denote the element of ${\mathbb{Z}}^V \oplus {\mathbb{Z}}^W$ with a 0 in the $v^{\text{th}}$ entry and $0$'s elsewhere.  If we consider $\begin{pmatrix} B^t-I \\ C^t \end{pmatrix} : {\mathbb{Z}}^V \rightarrow {\mathbb{Z}}^V \oplus {\mathbb{Z}}^W$, then $K_0(C^*(E)) \cong {\operatorname{coker}}\begin{pmatrix} B^t-I \\ C^t \end{pmatrix}$ via an isomorphism which takes $[p_v]_0$ to $[\delta_v]$ for each $v \in E^0$. Furthermore, this isomorphism takes $(K_0(C^*(E)))^+$ 