\chapter{Grassmanian}


The Grassmanian allows us to understand and give meaning to the following question;  What does it mean to change subspaces "smoothly"

\begin{definition}[Grassmanian]
smooth manifold formed from $k$-dimensional linear subspaces of $V$
$\mathbf{Gr}(k,V)$
\end{definition}


\begin{proposition}
	$\mathbf{Gr}(k,\mathbb{R}^n)$ is a compact smooth manifold of dimension $k(n-k)$
\end{proposition}