Proving that $\text{GL}(n,\mathbb{R})$ is open at $\text{gl}(n,\mathbb{R})$ and that its tangent space is isomorphic to $\text{gl}(n,\mathbb{R})$

Question

Hello. The image corresponds to chapter 8 of vector fields of the book introduction to the smooth manifolds of the author Lee.

Question 1. Why $\text{GL}(n,\mathbb{R})$ is an open set of the vector space $\text{gl}(n,\mathbb{R})$?

Question 2. Why its tangent space is naturally isomorphic to $\text{gl}(n,\mathbb{R})$?