  Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb
F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear
transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to
be similar if there exists a linear isomorphism $S:V\to V$ with
$SW=\widetilde{W}$ such that $S\circ T=\widetilde{T}\circ S $. Given a linear
map $T$ defined on a subspace $W$ of $V$, we give an explicit formula for the
number of linear maps that are similar to $T$. Our results extend a theorem of
Philip Hall that settles the case $W=V$ where the above problem is equivalent
to counting the number of square matrices over ${\mathbb F}_q$ in a conjugacy
class.
