  A weighted mean matrix whose weight sequence is linear with positive
coefficients is shown to be a posinormal operator on $\ell^2$. This operator is
also shown to be coposinormal, so it and its adjoint have the same null space
and the same range. The posinormality result leads to a proof that the weighted
mean matrix associated with the sequence of odd positive integers is
hyponormal, as well as a conjecture regarding a more general linear case.
