This problem is from "An Introduction to Manifolds" by Loring W. Tu.
I think I understand the solution, but I would like to ask if the following idea of approaching it is right.
When I dealt with this problem, I tried this way. Suppose $X_{I} \in T_{I}(GL(n,\mathbb{R}))$ and then we have $$\begin{align*} d(lg)_{I}(X_{I}) \in T_{g}(GL(n,\mathbb{R})) \end{align*}$$ Then, since $lg$ is a global diffeomorphism on $GL(n,\mathbb{R})$ , we have $$\begin{align*} d(lg)_{I}(X_{I}) = (lg_{*}X)_{g} \end{align*}$$ That is, the $X_{I}$ after the tangent map is just the pushforward of $X$ at $g$ . Then suppose $X_{I}$ has integral curve $\gamma$ , then $d(lg)_{I}(X_{I})$ should have integral curve $lg \circ \gamma$ which is $g \gamma$
The problem asks me to prove $$\begin{align*} gX_{I}= d(lg)_{I}(X_{I}) \end{align*}$$Thus I considered to evaluate them at the same smooth function $f$ $$\begin{align*} gX_{I}(f) &= g \frac{d}{dt} \bigg|_{t=0} (f \circ \gamma) (t) \\ &= g \frac{d}{dt} \bigg|_{t=0}f(\gamma(t)) \end{align*}$$ and $$\begin{align*} d(lg)_{I}(X_{I})(f)&= X_{I}(f \circ lg) \\ &= \frac{d}{dt} \bigg|_{t=0} f(g \gamma(t)) \end{align*}$$ Then I got stuck here. I think probably something goes wrong with my attempt. Any help on this? Thanks!