  We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V.
Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp.~971--1004] and, by viewing
matrices as coefficients for bivariate polynomials, we provide concise proofs
for key properties of linearizations for matrix polynomials. We also show that
every pencil in the double ansatz space is intrinsically connected to a
B\'{e}zout matrix, which we use to prove the eigenvalue exclusion theorem. In
addition our exposition allows for any polynomial basis and for any field. The
new viewpoint also leads to new results. We generalize the double ansatz space
by exploiting its algebraic interpretation as a space of B\'{e}zout pencils to
derive new linearizations with potential applications in the theory of
structured matrix polynomials. Moreover, we analyze the conditioning of double
ansatz space linearizations in the important practical case of a Chebyshev
basis.
