Notation: $\chi(G)$ is the set of smooth vector fields on Lie group $G$, which in fact forms a vector space.
Given a Lie group $G$, show that there exists a smooth vector field $X\in \chi(G)$, such that $X(g)\neq 0, \ \forall \ g\in G.$
Let $G$ be $n-$ dimensional Lie group. Show that there exists smooth vector fields $X_1, X_2,...,X_n$ such that for every $g\in G,$ the vectors $X_1(g), X_2(g),...,X_n(g)$ span the tangent space $T_g(G).$
I have no idea how to get started. Could you provide some hints?
Hint: Pick a vector $v\in T_{e}(G)$ - that is, pick a vector tangent to $G$ at the identity $e$. Now define a vector field $V$ on $G$ by the rule:
$V_g=(L_g)_{*}(v)$.
Here, $L_g:G\to G$ is the diffeomorphism of $G$ induced by left multiplication by $g$: $L_g(x)=gx$ for all $x\in G$. And, $(L_g)_{*}: T_e(G)\to T_{g}(G)$ is the induced differential. (Thus, $V_g\in T_{g}(G)$ for all $g\in G$.)
Exercise 1: Prove that $V$ is a smooth vector field on $G$ - that is, $V\in \chi(G)$.
Exercise 2: Prove that if $v_1,\dots ,v_n$ constitutes a basis of $T_{e}(G)$, then $V_1,\dots ,V_n$ has the property of $2$ in your question.
Exercise 3: Prove that $1$ in your question is a corollary of $2$.
Hope this helps! Please let me know if you need further hints.