Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given.
Also, I calculated the following regarding the Bruhat decomposition of $\operatorname{GL}_3$; can someone familiar with the subject tell me if it's correct?
$B(1 3)B \cong (\mathbb{A}^1\setminus \{0\})^3 \times \mathbb{A}^6$, cut out of $\operatorname{GL}_3$ by $a_{31} \neq 0$, $\left| \begin{smallmatrix} a_{21} & a_{22} \\\\ a_{31} & a_{32} \end{smallmatrix}\right| \neq 0$.
$B(1 2 3)B \cong (\mathbb{A}^1\setminus \{0\})^3 \times \mathbb{A}^5$, cut out of $\operatorname{GL}_3$ by $a_{31} \neq 0$, $\left| \begin{smallmatrix} a_{21} & a_{22} \\\\ a_{31} & a_{32} \end{smallmatrix}\right| = 0$.
$B(1 3 2)B \cong (\mathbb{A}^1\setminus \{0\})^3 \times \mathbb{A}^5$, cut out of $\operatorname{GL}_3$ by $a_{31} = 0$, $a_{21} \neq 0$, $a_{32} \neq 0$.
$B(1 2)B \cong (\mathbb{A}^1\setminus \{0\})^3 \times \mathbb{A}^4$, cut out of $\operatorname{GL}_3$ by $a_{31} = 0$, $a_{21} \neq 0$, $a_{32} = 0$
$B(2 3)B \cong (\mathbb{A}^1\setminus \{0\})^3 \times \mathbb{A}^4$, cut out of $\operatorname{GL}_3$ by $a_{31} = 0$, $a_{21} = 0$, $a_{32} \neq 0$
$B \cong (\mathbb{A}^1\setminus \{0\})^3 \times \mathbb{A}^3$, cut out of $\operatorname{GL}_3$ by $a_{21} = a_{31} = a_{32} = 0$.