 Solution

Suppose $V=L$. If $x\in L_{{\omega }_1}$, then $x\in L_{\alpha }$ for some countable ordinal $\alpha$. The set $L_{\alpha }$ is transitive and countable, so the transitive closure of $x$ lies in a countable set. Thus $x$ is hereditarily countable, proving

Conversely, let $x\in H_{{\aleph }_1}$ and choose a sufficiently large $L_{\theta }$ containing $x$. By the Downward Lowenheim-Skolem theorem, there is a countable elementary substructure $N\prec L_{\theta }$ that contains every member of $\mathrm{TC}(\{x\})$. The Mostowski collapse theorem gives a transitive collapse of $N$, and the condensation lemma for the constructible universe identifies it with $L_{\beta }$ for a countable ordinal $\beta$. Because $N$ contains the transitive closure of $x$ pointwise, the collapse fixes $x$. Thus $x\in L_{\beta }\subseteq L_{{\omega }_1}$. Hence