I don't like his proof that a 1-2 hidden unit system can't handle the red circle in the blue torus. Feels too handwavy.
My proof: Suppose the matrix W is singular. Then there exists a nonzero vector v for which W v = 0. Find a point x in the red circle, then find a different point y=x+av (for some scalar a) in the blue torus. Wy+b=W(x+av)+b=Wx+aWv+b=Wx+a0+b=wX+b. Fin.
My proof: Suppose the matrix W is singular. Then there exists a nonzero vector v for which W v = 0. Find a point x in the red circle, then find a different point y=x+av (for some scalar a) in the blue torus. Wy+b=W(x+av)+b=Wx+aWv+b=Wx+a0+b=wX+b. Fin.