Given a Lie group $G$, a vector field $X$ on $G$ is said to be left-invariant if \begin{align*} \left(\ell_g \right)_{*,h}(X_h) = X_{gh} \end{align*} for all $g,h\in G$, where $\ell_g:G\to G$, $\ell_g(x)=gx$, is the left multiplication by $g$. A point that is worth noting is that there exists a $1-1$ correspondence between the tangent space $T_{e}G$ of $G$ at the identity $e$ and $L(G)$, the vector space of all left-invariant vector fields on $G$, given by \begin{align*} (\phi(X_e))_g = (\ell_g)_{*,e}X_e. \end{align*}
Restricting our talk to the unit circle $S^1 \subset \mathbb{C}^\times$, I proved that the tangent space $T_{1}S^1$ can be identified with the vertical line $x=0$ so that each tangent vector $X_1 \in T_{1}S^1$ is a pair $(0,v)$ for some $v\in \mathbb{R}$, and that the tangent space $T_{z_0}S^1$ for any $z_0\in S^1$ is identified with the left-multiplication of the vertical line $x=0$ by $z_0$; that is, \begin{align*} T_{z_0}S^1 = \left\lbrace z_0 .(0,v): v\in \mathbb{R} \right\rbrace. \end{align*} However, I failed in finding the left-invariant vector fields on the unit circle. Would the results I got help in accomplishing my purpose ?.
I need help please, I appreciate any help.