Let \(T\) be a bounded linear operator defined on a separable, infinite dimensional Banach space \(X\). If there is an \(x\in X\) for which \(\{T^nx\}_{n=0}^{\infty}\) is dense in \(X\), then \(x\) is a hypercyclic vector and \(T\) is a hypercyclic operator. An infinite dimensional, closed linear subspace, \(H \subseteq X\), is hypercyclic if every \(x\in H\) is hypercyclic. \textit{A. Montes-Rodriguez} [Mich. Math. J. 43, No.~3, 419--436 (1996; Zbl 0907.47023)] proved a sufficient condition for \(T\) to admit a hypercyclic subspace. The proof is technical. This work provides a more elementary proof of this basic result. The proof relies on properties of the algebra of bounded linear operators on \(X\).