Let V be a vector space of countably infinite dimension over a division ring, and \(G=GL(V)\) be the group of all invertible linear transformations on V. Suppose H is a subgroup of G of index \(<2^{\aleph_ 0}\). Then the author proves that \(G_{(X)}\leq H\leq G_{\{X\}}\) for some finite- dimensional subspace X of V, where \(G_{(X)}\) and \(G_{\{X\}}\) denote, respectively, the pointwise and setwise stabilizers of X. This is a linear analogue of a result on symmetric groups proved in [\textit{J. D. Dixon}, \textit{P. M. Neumann}, and \textit{S. Thomas}, Bull. Lond. Math. Soc. 18, 580-586 (1986)]. The proof is independent but uses similar techniques.