While reading manifold theory I stuck to this problem: $V$ be a vector space with \dim V<\infty over $\mathbb{R}$ and $GL(V)$ be the group of all linear isomorphisms of $V$ into itself. A basis $e_1,\dots,e_n$ for $V$ induces a bijection $GL_n(\mathbb{R}\longrightarrow GL(V),$ $[a^i_j]\mapsto (e_j\mapsto\sum_{i}a^i_je_i),$ making $GL(V)$ into a $\mathcal{C}^{\infty}$ manifold, which we denote temporarily by $GL(V)_e$.
If $GL(V)_u$ is the manifold structure induced from the basis $u_1,\dots,u_n$ then how would you show they are diffeomorphic? Also I am not clearly understanding about the manifold structure of $GL(V)$, will be pleased for detail reply.
Would you give me the maximal atlas on $GL(V)$, coordinate charts, and $\mathcal{C}^{\infty}$ compatible maps? I guess If I some how get the those maps say $\phi_e$ and $\phi_u$ then $\phi_e o\phi_u$ would be a map by change of basis matrix $u$ to $e$ which wil be the diffeomorphism?