Solution
= Solution
Take the <quiver with one loop>. Its <path algebra> has basis $1,a,a^2,\ldots$ and multiplication $a^ra^s=a^{r+s}$, so it is the <polynomial ring> $k[a]$. Hence
$$
\boxed{kQ_1\cong k[a]:\ \text{generated by one algebra element, but infinite-dimensional over }k.}
$$
The algebra generators need not form a spanning vector-space set: products produce the infinitely many independent paths.