The exponential map by one-parameter subgroup on linear Lie group
Solution 1:
The one-parameter subgroup corresponding to $X\in\mathfrak{gl}_n(\Bbb R)(=T_eGL_n(\Bbb R))$ is the map$$\begin{array}{rccc}\varphi_X\colon&\Bbb R&\longrightarrow&GL_n(\Bbb R)\\&t&\mapsto&\exp(tX).\end{array}$$It's clearly a group homomorphism and $\varphi_X'(0)=X$. The same thing works with $\Bbb C$.