If g: Rn Rn and detg1 (x) 0, prove that in some open set containing we can

Question:

If g: Rn → Rn and detg1 (x) ≠ 0, prove that in some open set containing we can write g = to gn 0 ∙ ∙ ∙ o g1, 0.., where is of the form gi(x) = (x1, ∙ ∙ ∙ Fi (x) , ∙ ∙ ∙ Xn), and T is a linear transformation. Show that we can write g = gn o ∙ ∙ ∙ 0g1 if and only if g1 (x) is a diagonal matrix.