Let \(\Gamma\subset \text{GL}(V)\) be a group of linear maps on a finite-dimensional real vector space \(V\). By analogy with the case \(\text{SL}(n,\mathbb{Z})\subset \text{GL}(n,\mathbb{R})\), a vector \(v\) is called \(\Gamma\)-irrational if \(0\) is a limit point of the orbit \(\Gamma v\subset V\). The main result in this paper gives rather general conditions on \(\Gamma\) that guarantee the orbit closure of a \(\Gamma\)-irrational vector is large. Precisely, assume that the identity component \(G\) of the Zariski closure of \(\Gamma\) is a semisimple Lie group, and let \(v\) be a \(\Gamma\)-irrational vector. Then there exists a non-zero vector \(u\) and a connected abelian subgroup \(H\subset G\) comprising semisimple elements such that the dimension of \(H\) is the real rank of \(G\) and \(0\in\overline{Hu}\subset\overline{\Gamma v}\). The results here unify and strengthen several other results in the literature, and provide inter alia a partial solution to a problem posed by \textit{G. Margulis} [in: Arnold, V. (ed.) et al., Mathematics: frontiers and perspectives. Providence, RI: American Mathematical Society (AMS), 161--174 (2000; Zbl 0952.22005)], by showing that if \(\Gamma\subset \text{GL}(n,\mathbb{Z})\) is a semigroup for which the corresponding \(G\) is semisimple and \(Q\)-irreducible on \(\mathbb{R}^n\), then any orbit of \(\Gamma\) is either finite or dense.