Let $X$ and $Y$ be manifolds and $f: X \rightarrow Y$ a smooth map. Define the notions critical point, critical value, regular value of $f$. Prove that if $y$ is a regular value of $f$, then $f^{-1}(y)$ (if non-empty) is a smooth manifold of $\operatorname{dimension} \operatorname{dim} X-\operatorname{dim} Y$.

[The Inverse Function Theorem may be assumed without proof if accurately stated.]

Let $M_{n}(\mathbb{R})$ be the set of all real $n \times n$ matrices and $\operatorname{SO}(n) \subset M_{n}(\mathbb{R})$ the group of all orthogonal matrices with determinant 1 . Show that $\mathrm{SO}(n)$ is a smooth manifold and find its dimension.

Show further that $\mathrm{SO}(n)$ is compact and that its tangent space at $A \in \operatorname{SO}(n)$ is given by all matrices $H$ such that $A H^{t}+H A^{t}=0$.