Math 6510 • Differentiable Manifolds

Identify the set of all \(n\times n\) matrices \(M(n)\) with real entries with
\(\mathbb{R}^{n^2}\) and let
\[
  SL_n(\mathbb{R})=\{M\in M(n)\mid \det(M)=1\}.
\]
\begin{enumerate}
\item Show that \(SL_n(\mathbb{R})\) is a submanifold of \(M(n)\).
\item Compute the tangent space \(T_I(SL_n(\mathbb{R}))\) at the identity matrix.
\end{enumerate}