Let $p , k$ be two strictly positive integers and $V \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $V ^ { \mathrm { T } } V = I _ { k }$. For all $W \in \mathscr { M } _ { p , k } ( \mathbb { R } )$, $P_{V,W}$ denotes the matrix of the projection onto $\operatorname{Im}(V)$ parallel to $\operatorname{Im}(W)^\perp$ (when $W^T V$ is invertible). Show that there exists a neighborhood $\mathscr { V }$ of $V$ in $\mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $W ^ { \mathrm { T } } V$ is invertible for all $W \in \mathscr { V }$ and the map $W \mapsto P _ { V , W }$ is continuous from $\mathscr { V }$ to $\mathscr { M } _ { p } ( \mathbb { R } )$.
Let $p , k$ be two strictly positive integers and $V \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $V ^ { \mathrm { T } } V = I _ { k }$. For all $W \in \mathscr { M } _ { p , k } ( \mathbb { R } )$, $P_{V,W}$ denotes the matrix of the projection onto $\operatorname{Im}(V)$ parallel to $\operatorname{Im}(W)^\perp$ (when $W^T V$ is invertible).
Show that there exists a neighborhood $\mathscr { V }$ of $V$ in $\mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $W ^ { \mathrm { T } } V$ is invertible for all $W \in \mathscr { V }$ and the map $W \mapsto P _ { V , W }$ is continuous from $\mathscr { V }$ to $\mathscr { M } _ { p } ( \mathbb { R } )$.