The action of a set \(\mathcal S\) of linear operators on a vector space \(\mathcal V\) is, by definition, \(k\)-fold transitive if given linearly independent vectors \(\{x_1,x_2,\dots,x_k\}\) and arbitrary vectors \(\{y_1,y_2,\dots,y_k\}\), there is a member \(A\) of \(\mathcal S\) with \(Ax_i=y_i\) for all \(i\). It is shown that if the action of a Lie algebra of complex matrices is two-fold transitive, then it is either \(gl_n(\mathbb C)\) or, if \(n>2\), the Lie subalgebra \(sl_n(\mathbb C)\). Transitive action is not sufficient to yield this conclusion. Infinite-dimensional analogues are also considered.