Suppose that G is a closed subgroup of GLn(R) and v ∈ $T_eG$. In particular, this means that v ∈ $M_n(\mathbb R) = gl_n(\mathbb R)$. Let X denote the left invariant vector field on $GL_n(\mathbb R)$ such that $X_e = v$. Prove that X defines a left-invariant vector field on G.
I know how to extend the vector field inside $GL_n(\mathbb R)$ by pushing forward via $L_g$. Why should it land in the Tangent Space of G?
The tangent space $T_gG$ can as a subset of $\left\{g\right\}\times M(n,{\bf R})$ can be described as $\left\{(g,v)\colon ge^{tv}\in G\ \forall t\in{\bf R}\right\}$. With this notation, $DL_g$ sends $(e,v)$ to $(g,v)$.
If $v\in T_e v$, then $e^{tv}\in G$ for all t, hence $ge^{tv}\in G$ for all t.
But this means exactly that $DL_g$ sends $T_eG$ to $T_gG$.