GL(n,R) is an open subset of the Euclidian n^2-space

ga/13lie_groups.tex

@@ -27,7 +27,7 @@ \subsection{Lie groups}
 
 \item Let $S^1:=\{z\in\C\mid |z|=1\}$ and let $\cdot$ be the usual multiplication of complex numbers. Then $(S^1,\cdot)$ is a commutative Lie group usually denoted $\mathrm{U}(1)$.
 
-\item Let $\mathrm{GL}(n,\R)=\{\phi\cl\R^n\xrightarrow{\sim}\R^n\mid \det \phi \neq 0\}$. This set can be endowed with the structure of a smooth $n^2$-dimensional manifold, by noting that there is a bijection between linear maps $\phi\cl\R^n\xrightarrow{\sim}\R^n$ and $\R^{2n}$. The condition $\det \phi\neq 0$ is a so-called \emph{open condition}, meaning that $\mathrm{GL}(n,\R)$ can be identified with an open subset of $\R^{2n}$, from which it then inherits a smooth structure.
+\item Let $\mathrm{GL}(n,\R)=\{\phi\cl\R^n\xrightarrow{\sim}\R^n\mid \det \phi \neq 0\}$. This set can be endowed with the structure of a smooth $n^2$-dimensional manifold, by noting that there is a bijection between linear maps $\phi\cl\R^n\xrightarrow{\sim}\R^n$ and $\R^{n^2}$. The condition $\det \phi\neq 0$ is a so-called \emph{open condition}, meaning that $\mathrm{GL}(n,\R)$ can be identified with an open subset of $\R^{n^2}$, from which it then inherits a smooth structure.
 
 Then, $(\mathrm{GL}(n,\R),\circ)$ is a Lie group called the \emph{general linear group}.