How can we show that $lg_{∗I} : T_IGL(n, \Bbb R) → T_gGL(n, \Bbb R)$ is given by $v \to gv$?

Question

Let $g \in GL(n, \Bbb R)$ and $l_g : GL(n, \Bbb R) → GL(n, \Bbb R)$ be given by $l_g(h) = gh$. How can we show that $lg_{∗I} : T_IGL(n, \Bbb R) → T_gGL(n, \Bbb R)$ is given by $v \to gv$?

Answer 1

$\frac{d}{dt}g(I+tv)|_{t=0} =gv$