  We consider the capability of $p$-groups of class two and odd prime exponent.
The question of capability is shown to be equivalent to a statement about
vector spaces and linear transformations, and using the equivalence we give
proofs of some old results and several new ones. In particular, we establish a
number of new necessary and new sufficient conditions for capability, including
a sufficient condition based only on the ranks of $G/Z(G)$ and $[G,G]$.
Finally, we characterise the capable groups among the 5-generated groups in
this class.
