  In this paper, we consider the space $\mathrm{BV}^{\mathbb A}(\Omega)$ of
functions of bounded $\mathbb A$-variation. For a given first order linear
homogeneous differential operator with constant coefficients $\mathbb A$, this
is the space of $L^1$--functions $u:\Omega\rightarrow\mathbb R^N$ such that the
distributional differential expression $\mathbb A u$ is a finite (vectorial)
Radon measure. We show that for Lipschitz domains $\Omega\subset\mathbb R^{n}$,
$\mathrm{BV}^{\mathbb A}(\Omega)$-functions have an $L^1(\partial\Omega)$-trace
if and only if $\mathbb A$ is $\mathbb C$-elliptic (or, equivalently, if the
kernel of $\mathbb A$ is finite dimensional). The existence of an
$L^1(\partial\Omega)$-trace was previously only known for the special cases
that $\mathbb A u$ coincides either with the full or the symmetric gradient of
the function $u$ (and hence covered the special cases $\mathrm{BV}$ or
$\mathrm{BD}$). As a main novelty, we do not use the fundamental theorem of
calculus to construct the trace operator (an approach which is only available
in the $\mathrm{BV}$- and $\mathrm{BD}$-setting) but rather compare projections
onto the nullspace as we approach the boundary. As a sample application, we
study the Dirichlet problem for quasiconvex variational functionals with linear
growth depending on $\mathbb A u$.
