Let $G$ be a group which contains a linear subgroup of finite index. Is $G$ necessarily linear? What if $G$ is finitely generated?
I cannot seem to find anything which talks about "virtually linear" groups, and so I would hypothesise that "virtually linear $\Rightarrow$ linear". Also, many properties of linear groups are virtual properties, such as the Tits alternative and residually finite. However, these facts do not constitute a proof!