Let $G=GL(n,\mathbb R)$, show that this application $ (A,B) \in G \times G \rightarrow AB \in G$ is $C^{\infty}$

Question

Let be $G=GL(n,\mathbb R)$.

I consider the application $$a: G \times G \rightarrow G$$ such that $$ (A,B) \rightarrow AB .$$

I have to prove that this application is $C^\infty$.

I know the definition of differentiability, but I'm stuck because I cannot find explicity a chart on $G$.

Any hint would be greatly appreciated.

Answer 1

An explicit chart sends $GL_n(\mathbb{R}) \to \mathbb{R}^{n^2}$ taking a matrix to its $n^2$ coefficients strung out in a row vector.

To see that multiplication is smooth, check that when you use this chart, The map is just polynomial equations in the coordinates. Since polynomials are smooth, so is matrix multiplication!