$GL(n)$ is the space of invertible matrices.
Let $\phi^i_j: GL(n) \rightarrow R$ be the function where $\phi^i_j(g)$ is the element in the $i$-th row and $j$-th column of $g$.
Let $d\phi$ be the matrix of $1$-forms, where the element in the $i$-th row and $j$-th column is $d\phi^i_j$.
Let $F: GL(n) \rightarrow GL(n):g \mapsto g^{-1}.$
For each $h \in GL(n)$, there is a map, called left translation, $L_h: GL(n) \rightarrow GL(n): g \mapsto hg.$
Let $\Theta$ be the matrix of $1$-forms given by $\Theta(g) = F(g) d\phi.$
I need to compute $L_h^*\Theta$ and the derivative of $\Theta$.
But I have no idea how to solve these questions because I am not familiar with these operations on matrices.
Could you please help me? Thanks a lot.
Note that $L_h$ is the restriction to $G$ of a linear map on $M(n,R)$ $h, M\to g M$, so that $L_h^* d\Phi= h d\Phi$, as $F(hg)=F(g).h^{-1}$, we see that $L_h^*\Theta= \Theta$