space spans only the nonredundant part of the redundant internal coordinates. The constrained and unconstrained nonredundant spaces together also span the nonredundant space. The redundant internal coordinates can be projected into the nonredundant space, the constrained space, and the unconstrained space with the N, P, and Q matrices respectively. The nonredundant internal coordinates can be projected into the constrained and unconstrained nonredundant space with the P and Q matrices respectively. Note that when performing an unconstrained optimization, the nonredundant and unconstrained nonredundant spaces are identical and the constrained space is zerodimensional.