+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
+ sage: lls = L.vector_space().span(ll_basis)
+ sage: Z_cone.linear_subspace() == lls
+ True
+
+ The lineality of the Z-transformations on a cone is the Lyapunov
+ rank of that cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_cone.lineality() == K.lyapunov_rank()
+ True
+
+ The lineality spaces of the duals of the positive operator and
+ Z-transformation cones are equal. From this it follows that the
+ dimensions of the Z-transformation cone and positive operator cone
+ are equal::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_cone.dim() == Z_cone.dim()
+ True
+ sage: pi_star = pi_cone.dual()
+ sage: z_star = Z_cone.dual()
+ sage: pi_star.linear_subspace() == z_star.linear_subspace()
+ True
+
+ The trivial cone, full space, and half-plane all give rise to the
+ expected dimensions::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False)
+ sage: Z_cone.dim() == 3