def lift(self, x):
r"""
Given an element of the image, return an element of the codomain that maps onto it.
Note that ``lift`` and ``preimage_representative`` are
equivalent names for this method, with the latter suggesting
that the return value is a coset representative of the domain
modulo the kernel of the morphism.
EXAMPLE::
sage: X = QQ**2
sage: V = X.span([[2, 0], [0, 8]], ZZ)
sage: W = (QQ**1).span([[1/12]], ZZ)
sage: f = V.hom([W([1/3]), W([1/2])], W)
sage: f.lift([1/3])
(8, -16)
sage: f.lift([1/2])
(12, -24)
sage: f.lift([1/6])
(4, -8)
sage: f.lift([1/12])
Traceback (most recent call last):
...
ValueError: element is not in the image
sage: f.lift([1/24])
Traceback (most recent call last):
...
TypeError: element (= [1/24]) is not in free module
This works for vector spaces, too::
sage: V = VectorSpace(GF(3), 2)
sage: W = VectorSpace(GF(3), 3)
sage: f = V.hom([W.1, W.1 - W.0])
sage: f.lift(W.1)
(1, 0)
sage: f.lift(W.2)
Traceback (most recent call last):
...
ValueError: element is not in the image
sage: w = W((17, -2, 0))
sage: f(f.lift(w)) == w
True
This example illustrates the use of the ``preimage_representative``
as an equivalent name for this method. ::
sage: V = ZZ^3
sage: W = ZZ^2
sage: w = vector(ZZ, [1,2])
sage: f = V.hom([w, w, w], W)
sage: f.preimage_representative(vector(ZZ, [10, 20]))
(0, 0, 10)
"""
from free_module_element import vector
x = self.codomain()(x)
A = self.matrix()
R = self.base_ring()
if R.is_field():
try:
C = A.solve_left(x)
except ValueError:
raise ValueError("element is not in the image")
else:
# see inverse_image for similar code but with comments
if not hasattr(A, 'hermite_form'):
raise NotImplementedError("base ring (%s) must have hermite_form algorithm in order to compute inverse image"%R)
H, U = A.hermite_form(transformation=True,include_zero_rows=False)
Y = H.solve_left(vector(self.codomain().coordinates(x)))
C = Y*U
try:
t = self.domain().linear_combination_of_basis(C)
except TypeError:
raise ValueError("element is not in the image")
assert self(t) == x
return t
def lift(self, x):
r"""
Given an element of the image, return an element of the codomain that maps onto it.
Note that ``lift`` and ``preimage_representative`` are
equivalent names for this method, with the latter suggesting
that the return value is a coset representative of the domain
modulo the kernel of the morphism.
EXAMPLE::
sage: X = QQ**2
sage: V = X.span([[2, 0], [0, 8]], ZZ)
sage: W = (QQ**1).span([[1/12]], ZZ)
sage: f = V.hom([W([1/3]), W([1/2])], W)
sage: f.lift([1/3])
(8, -16)
sage: f.lift([1/2])
(12, -24)
sage: f.lift([1/6])
(4, -8)
sage: f.lift([1/12])
Traceback (most recent call last):
...
ValueError: element is not in the image
sage: f.lift([1/24])
Traceback (most recent call last):
...
TypeError: element [1/24] is not in free module
This works for vector spaces, too::
sage: V = VectorSpace(GF(3), 2)
sage: W = VectorSpace(GF(3), 3)
sage: f = V.hom([W.1, W.1 - W.0])
sage: f.lift(W.1)
(1, 0)
sage: f.lift(W.2)
Traceback (most recent call last):
...
ValueError: element is not in the image
sage: w = W((17, -2, 0))
sage: f(f.lift(w)) == w
True
This example illustrates the use of the ``preimage_representative``
as an equivalent name for this method. ::
sage: V = ZZ^3
sage: W = ZZ^2
sage: w = vector(ZZ, [1,2])
sage: f = V.hom([w, w, w], W)
sage: f.preimage_representative(vector(ZZ, [10, 20]))
(0, 0, 10)
"""
from free_module_element import vector
x = self.codomain()(x)
A = self.matrix()
R = self.base_ring()
if R.is_field():
try:
C = A.solve_left(x)
except ValueError:
raise ValueError("element is not in the image")
else:
# see inverse_image for similar code but with comments
if not hasattr(A, 'hermite_form'):
raise NotImplementedError(
"base ring (%s) must have hermite_form algorithm in order to compute inverse image"
% R)
H, U = A.hermite_form(transformation=True, include_zero_rows=False)
Y = H.solve_left(vector(self.codomain().coordinates(x)))
C = Y * U
try:
t = self.domain().linear_combination_of_basis(C)
except TypeError:
raise ValueError("element is not in the image")
assert self(t) == x
return t