  We give a complete description of the cohomology ring $A^*(\overline Z)$ of a
compactification of a linear subvariety $Z$ of a torus in a smooth toric
variety whose fan $\Sigma$ is supported on the tropicalization of $Z$. It turns
out that cocycles on $\overline Z$ canonically correspond to Minkowski weights
on $\Sigma$ and that the cup product is described by the intersection product
on the tropical matroid variety $\operatorname{Trop}(Z)$.
