Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples

For a subgroup $L$ of the symmetric group $S_{\mathrm{\ell }}$, we determine the minimal base size of $G{L}_d(q)\wr L$ acting on $V_d(q{)}^{\mathrm{\ell }}$ as an imprimitive linear group. This is achieved by computing the number of orbits of $G{L}_d(q)$ on spanning $m$-tuples, which turns out to be the number of $d$-dimensional subspaces of $V_m(q)$. We then use these results to prove that for certain families of subgroups $L$, the affine groups whose stabilisers are large subgroups of $G{L}_d(q)\wr L$ satisfy a conjecture of Pyber concerning bases.