Flow generated by a matrix Lie group

Question

I found this question in a book by Baez and Muniain on Gauge theory and knots. It was given in one of the exercises. $G$ is a matrix Lie group and $v$ is a left invariant vector field defined on $G$. $v_{1}$ is the value of the vector field at the identity element of $G$. Let $\phi_{t}$:$G\to G$ be given by $$\phi_{t}(g)=g \exp(tv_{1})$$. We have to show that $\phi_{t}$ is the flow generated by $v$, that is,that $$\frac{d\phi_{t}(g)}{dt}|_{t=0}=v_{g}$$ for all $g\in G$. What is the best way of proving this result?

Answer 1

Put very briefly (to the point of not doing justice to all the subtleties involved): the special case $g = 1$ has nothing to do with Lie groups, it is just another example where you can see the equivalence of the various definitions of tangent vector on a manifold at work.

Now if you understand the $g = 1$ case, then the fact that changing the $1$ into a $g$ both left and right of the $=$-sign doesn't invalidate the equality is, in essence, just the meaning of the word 'invariant' in the term 'invariant vector field'.