Satellite beam-switching problems are studied where there are m up-link beams, n down-link beams and multiple carriers per beam. By augmenting a traffic matrix with an extra row and column, it is possible to find a sequence of switching modes ((0, 1)-matrices with generally multiple unit entries in each row and column) that realize optimal transmission time. Switching modes generated are shown to be linearly independent. The number of switching modes required for an m x n matrix is bounded by (m - 1)(n - 1) + 1. For an augmented (m + 1) x (n + 1) matrix, the bound is then mn + 1. The bounds on the number of switching modes and the computational complexity for a number of related satellite transmission scheduling problems are lowered. In simplified form, the results (particularly the linear independence of permutation matrices generated) apply to algorithmic decomposition of doubly stochastic matrices into convex combinations of permutation matrices.