Lately I've been reading about linear programming and I just read a theorem about extreme directions in a polyhedral set, the problem is that this theorem has no proof and I can't figure how I can do the proof by myself. Here's the theorem:
Let $X=\{x: Ax \le b, \ x\ge 0 \}$ where $A$ is an $m\times n$ matrix with rank $m$. Then $d$ is an extreme direction of $X$ if and only if $d$ is a positive multiple of the vector $(-y^{t}_{j},0,0,...,1,0,0,...,0)^{t}$ where the $1$ appears in position j, and where:
$y_{j}=B^{-1}a_{j}\le 0$
$A = [B,N]$ where $B$ is an $m\times m$ invertible matrix
$a_{j}= $ a column of $N$
Hope someone could help me do the proof.