  The seminal work by Mackey et al. in 2006 (reference [21] of the article)
introduced vector spaces of matrix pencils, with the property that almost all
the pencils in the spaces are strong linearizations of a given square regular
matrix polynomial. This work was subsequently extended by De Ter\'an et al. in
2009 (reference [5] of the article) to include the case of square singular
matrix polynomials. We extend this work to non-square matrix polynomials by
proposing similar vector spaces of rectangular matrix pencils that are equal to
the ones introduced by Mackey et al. when the polynomial is square. Moreover,
the properties of these vector spaces are similar to those in the article by De
Ter\'an et al. for the singular case. In particular, the complete eigenvalue
problem associated with the matrix polynomial can be solved by using almost
every matrix pencil from these spaces. Further, almost every pencil in these
spaces can be trimmed to form many smaller pencils that are strong
linearizations of the matrix polynomial which readily solve the complete
eigenvalue problem for the polynomial. These linearizations are easier to
construct and are often smaller than the Fiedler linearizations introduced by
De Ter\'an et al. in 2012 (reference [7] of the article). Further, the global
backward error analysis by Dopico et al. in 2016 (reference [10] of the
article) applied to these linearizations, shows that they provide a wide choice
of linearizations with respect to which the complete polynomial eigenvalue
problem can be solved in a globally backward stable manner.
