Problem: Let \[ O(n,\mathbb{R} ) = \left\{ A\in M(n,\mathbb{R} ): AA ^T = I_n = A^{T}A \right\} \] be the set of $n\times n$ orthogonal matrices and \[ SO(n,\mathbb{R} ) = \left\{ A\in O(n,\mathbb{R}): \det(A) = 1 \right\} \] be the set of all $n\times n$ special orthogonal matrices. Show that $O(n,\mathbb{R} )$ is a compact set. Is it connected? Also, show that $SO(n,\mathbb{R})$ is closed and connected.