let $K$ be a field, $n \geq 1$. denote $E_{i,j}$ the elementary matrix having $1$ on the diagonale and in the entry $(i,j)$, and $E_i(a)$ the elementary matrix $diag(1,...,a,...,1)$. you know that $GL_n(K)$ is generated by these matrices, but what relations do we need in order to get a presentation for $GL_n(K)$?
 
here are some relations, which correspond to simple relations about row operations:
 
- $E_i(1)=1$
- $E_i(ab) = E_i(a) E_i(b)$
- $E_i(a) E_j(b) = E_j(b) E_i(a)$
- $(E_j(-1) E_{ij})^2=1$
- $E_j(a+b)^{-1} E_{ij} E_j(a+b) = E_j(a)^{-1} E_{ij} E_j(b) E_i(a)^{-1} E_{ij} E_j(a)$
- $ (E_{ji} E_{ij} E_{ji} E_j(-1))^2=1$
 
are these all relations? how can we prove that?