  A well-known conjecture of Babai states that if $G$ is a finite simple group
and $X$ is a generating set of $G$, then the diameter of the Cayley graph
$Cay(G,X)$ is bounded above by $(\log |G|)^c$ for some absolute constant $c$.
The goal of this paper is to prove such a bound for the diameter of $Cay(G,X)$
whenever $G=SL(n,p)$ and $X$ is a generating set of $G$ which contains a
transvection. A natural analogue of this result is also proved for $G=SL(n,K)$,
where $K$ can be any field.
