linear algebra - Operation of GL_n(Z/bZ)

Any transformation
$$(v_1,ldots,v_n)mapsto (v_1,ldots,v_{j-1},v_j+av_k,v_{j+1},ldots,v_n)$$
for $jne k$ is achievable by means of some such matrix. It suffices to reduce
an admissible vector to $(1,0,ldots,0)$ by means of a sequence of such reductions.
I would do it in three stages

1. Make $v_n$ into a unit in $mathbb{Z}/bmathbb{Z}$;




2. Make $v_1=1$;




3. Make all $v_j=0$ for $j>1$.