= Solution
Number the source vertices $1,\ldots,r$ and the sink $c=r+1$, and write $a_i:i\to c$ for the arrows. The only basis paths are $e_1,\ldots,e_r,e_c,a_1,\ldots,a_r$. Using standard <matrix units>, define
$$
e_i\longmapsto E_{ii},\qquad e_c\longmapsto E_{cc},\qquad a_i\longmapsto E_{ci}.
$$
The relations $e_ca_i=a_i=a_ie_i$, orthogonality of the vertex <idempotents>, and $a_ia_j=0$ agree exactly with multiplication of these <matrix units>. The map is therefore an algebra homomorphism, and its images form a basis of the algebra
$$
\boxed{kS_r\cong\left\{\begin{pmatrix}\operatorname{diag}(\lambda_1,\ldots,\lambda_r)&0\\v&\mu\end{pmatrix}:\lambda_i,\mu\in k,\ v\in k^{1\times r}\right\}.}
$$
This is an explicit isomorphism of the <path algebra> of the <subspace quiver> with a $(2r+1)$-dimensional triangular matrix algebra.
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