  This paper considers the Pontryagin characters of graded vector bundles of
finite rank, in the cohomology vector spaces of a Lie algebroid over the same
base. These Pontryagin characters vanish if the graded vector bundle carries a
representation up to homotopy of the Lie algebroid. As a consequence, this
gives a strong obstruction to the existence of a representation up to homotopy
on a graded vector bundle of finite rank. In particular, if a graded vector
bundle $E[0]\oplus F[1]\to M$ carries a $2$-term representation up to homotopy
of a Lie algebroid $A\to M$, then all the (classical) $A$-Pontryagin classes of
$E$ and $F$ must coincide. This paper generalises as well Bott's vanishing
theorem to the setting of Lie algebroid representations (up to homotopy) on
arbitrary vector bundles. As an application, the main theorems induce new
obstructions to the existence of infinitesimal ideal systems in a given Lie
algebroid.
