Does this subset of $GL_n(\mathbb R)$ have two connected components or more?

Question

Let $\{v_1, \dots, v_r\}$ be a fixed linearly independent set in $\mathbb R^n$ with $1 \le r < n$. Let us define a set \begin{align*} \mathcal E = \{V \in GL_n(\mathbb R): V = (v_1, \dots, v_r, *, \dots, *)\}, \end{align*} where $*$ means that column can assume any vector in $\mathbb R^n$. Certainly, $*$ should not be in the span of $\{v_1, \dots, v_r\}$. I am interested in determining whether $\mathcal E$ has two connected components (or more).

Answer 1

Extend $\{v_1,\ldots,v_r\}$ to a full basis and let $W=\pmatrix{v_1&\cdots&v_n}$. Then every $V\in\mathcal E$ can be written as $V=W\pmatrix{I_r&R\\ 0&S}$ where $S\in GL_{n-r}(\mathbb R)$ and $R\in M_{r\times(n-r)}(\mathbb R)$. Therefore $\mathcal E$ has two connected components, which are essentially $M_{r\times(n-r)}(\mathbb R) \times GL_{n-r}^+(\mathbb R)$ and $M_{r\times(n-r)}(\mathbb R) \times GL_{n-r}^-(\mathbb R)$.