# Triangulability theorem

Let $k$ be an algebraically closed field. Denote by $GL_n(k)$ the general linear group of invertible $n \times n$ matrices over $k$ and by $B_n(k)$ the Borel subgroup: the subgroup of invertible upper triangular matrices. Then, $B(n,k)$ is conjugate-dense in $GL(n,k)$. In other words, given any matrix in $GL(n,k)$, we can conjugate it (or change basis) to make it upper triangular.
Note that the result fails over non-algebraically closed fields; in particular, it fails when $k$ is a finite field.