This is not true. Consider with basis . The space spammed by is the x-axis,y-axis, z-axis respectively. The space spammed by is the xy-plane, xz-plane, yz-plane respectively. Therefore, any subspace that is distinct from these will not be spammed by a subset of this basis. For example, the plane is a subspace but it one one of those coordinate planes and so it cannot be spammed by a subset of the basis.