sentences:  If the columns of the matrix formula_47 are an orthonormal basis for the null space of
  and the columns of the matrix formula_49 are an orthonormal basis for the left null space of formula_50, then
  the system formula_51
  can be rewritten as
  where formula_53 is in the complement of the null space of formula_50 formula_55.In the first equation, which is parametrized by the null space of the Jacobian (formula_56), the Jacobian with respect to formula_53 is non-singular.So the implicit function theorem states that there is a mapping formula_58 such that formula_59 and formula_60.