I read somewhere that given an invertible matrix, we can decompose it into a product of lower triangular and upper triangular matrices up to a permutation matrix using the Gauss elimination method.
So we have $GL_n(\mathbb C)=\cup_{w \in S_n}BwB$, where $B$ is the set of all upper triangular matrices.
Could someone please explain with an example (say with a $4 \times 4$ matrix) how the permutation matrix come to the picture (possibly the elementary rwo operations corresponding to the permutation of rows) and how the dimension of $BwB/B$ is $l(w)$, the length of the $w$.