  Time-series of persistence diagrams, known as vineyards, have shown to be
useful in diverse applications. A natural algebraic version of vineyards is a
time series of persistence modules equipped with interleaving maps between the
persistence modules at different time values. We call this a vineyard module.
In this paper we will set up the framework for representing vineyards modules
via families of matrices and outline an algorithmic way to change the bases of
the persistence modules at each time step within the vineyard module to make
the matrices in this representation as simple as possible. With some reasonable
assumptions on the vineyard modules, this simplified representation of the
vineyard module can be completely described (up to isomorphism) by the
underlying vineyard and a vector of finite length. We first must set up a lot
of preliminary results about changes of bases for persistence modules where we
are given $\epsilon$-interleaving maps for sufficiently close $\epsilon$. While
this vector representation is not in general guaranteed to be unique we can
prove that it will be always zero when the vineyard module is isomorphic to the
direct sum of vine modules. This new perspective on vineyards provides an
interesting and yet tractable case study within multi-parameter persistence.
