Interpolation Between Bases and the Shuffle Exchange Network

Let u₁. . . , u_n and v₁, . . . , v_n be bases of a vector space (the interesting case, when the underlying field is finite). Then there exist vectors w₁, . . . , w_n-1, such that every n consecutive vectors in the sequence u₁, . . . , u_n, W₁, . . . , W_n-l, v₁, . . . , v_n from a basis. Similar statements hold in structures other then vector spaces. The case of a free Boolean algebra is shown equivalent to an open problem in switching network theory.