Let $n\in\mathbb N$, let $\mathbb F\in\{\mathbb R,\mathbb C\}$, and view $GL(n,\mathbb F)$ as a real Lie group with its usual smooth manifold structure as an open subset of the real [vector space](/page/Vector%20Space) $M(n,\mathbb F)$. For $A\in M(n,\mathbb F)$, let $\exp(A)$ denote the matrix exponential defined by the absolutely convergent series

\begin{align*} \exp(A)=\sum_{m=0}^{\infty}\frac{A^m}{m!}. \end{align*}

For every $X\in M(n,\mathbb F)$, the map

\begin{align*} \gamma_X:\mathbb R\to GL(n,\mathbb F),\qquad \gamma_X(t)=\exp(tX) \end{align*}

is a smooth [group homomorphism](/page/Group%20Homomorphism) from $(\mathbb R,+)$ to $GL(n,\mathbb F)$.