Let $n\in\mathbb N$ and let $\mathbb F\in\{\mathbb R,\mathbb C\}$. Equip $M(n,\mathbb F)$ with its usual Euclidean topology, regarding $M(n,\mathbb C)$ as a real [vector space](/page/Vector%20Space) of dimension $2n^2$. Then
is an open subset of $M(n,\mathbb F)$. With the smooth manifold structure inherited from this open subset, $GL(n,\mathbb F)$ is a real Lie group under matrix multiplication.