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r"""

Homspaces between free modules

EXAMPLES: We create `\mathrm{End}(\ZZ^2)` and compute a

basis.

sage: M = FreeModule(IntegerRing(),2)

sage: E = End(M)

sage: B = E.basis()

sage: len(B)

sage: B[0]

Free module morphism defined by the matrix

[1 0]

[0 0]

Domain: Ambient free module of rank 2 over the principal ideal domain ...

Codomain: Ambient free module of rank 2 over the principal ideal domain ...

We create `\mathrm{Hom}(\ZZ^3, \ZZ^2)` and

compute a basis.

sage: V3 = FreeModule(IntegerRing(),3)

sage: V2 = FreeModule(IntegerRing(),2)

sage: H = Hom(V3,V2)

sage: H

Set of Morphisms from Ambient free module of rank 3 over

the principal ideal domain Integer Ring

to Ambient free module of rank 2

over the principal ideal domain Integer Ring

in Category of finite dimensional modules with basis over

(euclidean domains and infinite enumerated sets and metric spaces)

sage: B = H.basis()

sage: len(B)

sage: B[0]

Free module morphism defined by the matrix

[1 0]

[0 0]

[0 0]...

TESTS::

sage: H = Hom(QQ^2, QQ^1)

sage: loads(dumps(H)) == H

True

See :trac:`5886`::

sage: V = (ZZ^2).span_of_basis([[1,2],[3,4]])

sage: V.hom([V.0, V.1])

Free module morphism defined by the matrix

[1 0]

[0 1]...

"""

#*****************************************************************************

# Distributed under the terms of the GNU General Public License (GPL)

# This code is distributed in the hope that it will be useful,

# but WITHOUT ANY WARRANTY; without even the implied warranty

# of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

# See the GNU General Public License for more details; the full text

# is available at:

# http://www.gnu.org/licenses/

#*****************************************************************************

from __future__ import absolute_import

import sage.categories.homset

from sage.structure.element import is_Matrix

from sage.matrix.constructor import matrix, identity_matrix

from sage.matrix.matrix_space import MatrixSpace

from inspect import isfunction

from sage.misc.cachefunc import cached_method

def is_FreeModuleHomspace(x):

r"""

Return ``True`` if ``x`` is a free module homspace.

EXAMPLES:

Notice that every vector space is a free module, but when we construct

a set of morphisms between two vector spaces, it is a

``VectorSpaceHomspace``, which qualifies as a ``FreeModuleHomspace``,

since the former is special case of the latter.

sage: H = Hom(ZZ^3, ZZ^2)

sage: type(H)

sage: sage.modules.free_module_homspace.is_FreeModuleHomspace(H)

True

sage: K = Hom(QQ^3, ZZ^2)

sage: type(K)

sage: sage.modules.free_module_homspace.is_FreeModuleHomspace(K)

True

sage: L = Hom(ZZ^3, QQ^2)

sage: type(L)

sage: sage.modules.free_module_homspace.is_FreeModuleHomspace(L)

True

sage: P = Hom(QQ^3, QQ^2)

sage: type(P)

sage: sage.modules.free_module_homspace.is_FreeModuleHomspace(P)

True

sage: sage.modules.free_module_homspace.is_FreeModuleHomspace('junk')

False

"""

return isinstance(x, FreeModuleHomspace)

class FreeModuleHomspace(sage.categories.homset.HomsetWithBase):

def __call__(self, A, check=True):

r"""

INPUT:

- A -- either a matrix or a list/tuple of images of generators,

or a function returning elements of the codomain for elements

of the domain.

- check -- bool (default: True)

If A is a matrix, then it is the matrix of this linear

transformation, with respect to the basis for the domain and

codomain. Thus the identity matrix always defines the

identity morphism.

EXAMPLES::

sage: V = (ZZ^3).span_of_basis([[1,1,0],[1,0,2]])

sage: H = V.Hom(V); H

Set of Morphisms from ...

sage: H([V.0,V.1]) # indirect doctest

Free module morphism defined by the matrix

[1 0]

[0 1]...

sage: phi = H([V.1,V.0]); phi

Free module morphism defined by the matrix

[0 1]

[1 0]...

sage: phi(V.1) == V.0

True

sage: phi(V.0) == V.1

True

The following tests against a bug that was fixed in

:trac:`9944`. The method ``zero()`` calls this hom space with

a function, not with a matrix, and that case had previously

not been taken care of::

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ)

sage: V.Hom(V).zero() # indirect doctest

Free module morphism defined by the matrix

[0 0 0]

Domain: Free module of degree 3 and rank 3 over Integer Ring

Echelon ...

Codomain: Free module of degree 3 and rank 3 over Integer Ring

Echelon ...

"""

from . import free_module_morphism

if not is_Matrix(A):

# Compute the matrix of the morphism that sends the

# generators of the domain to the elements of A.

C = self.codomain()

try:

if isfunction(A):

v = [C(A(g)) for g in self.domain().gens()]

else:

v = [C(a) for a in A]

A = matrix([C.coordinates(a) for a in v], ncols=C.rank())

except TypeError:

# Let us hope that FreeModuleMorphism knows to handle

# that case

pass

return free_module_morphism.FreeModuleMorphism(self, A)

@cached_method

def zero(self):

"""

EXAMPLES::

sage: E = ZZ^2

sage: F = ZZ^3

sage: H = Hom(E, F)

sage: f = H.zero()

sage: f

Free module morphism defined by the matrix

[0 0 0]

Domain: Ambient free module of rank 2 over the principal ideal domain Integer Ring

Codomain: Ambient free module of rank 3 over the principal ideal domain Integer Ring

sage: f(E.an_element())

(0, 0, 0)

sage: f(E.an_element()) == F.zero()

True

TESTS:

We check that ``H.zero()`` is picklable::

sage: loads(dumps(f.parent().zero()))

Free module morphism defined by the matrix

[0 0 0]

Domain: Ambient free module of rank 2 over the principal ideal domain Integer Ring

Codomain: Ambient free module of rank 3 over the principal ideal domain Integer Ring

"""

return self(lambda x: self.codomain().zero())

def _matrix_space(self):

"""

Return underlying matrix space that contains the matrices that define

the homomorphisms in this free module homspace.

OUTPUT:

- matrix space

EXAMPLES::

sage: H = Hom(QQ^3, QQ^2)

sage: H._matrix_space()

Full MatrixSpace of 3 by 2 dense matrices over Rational Field

"""

try:

return self.__matrix_space

except AttributeError:

R = self.codomain().base_ring()

M = MatrixSpace(R, self.domain().rank(), self.codomain().rank())

self.__matrix_space = M

return M

def basis(self):

"""

Return a basis for this space of free module homomorphisms.

OUTPUT:

- tuple

EXAMPLES::

sage: H = Hom(ZZ^2, ZZ^1)

sage: H.basis()

(Free module morphism defined by the matrix

[1]

[0]

Domain: Ambient free module of rank 2 over the principal ideal domain ...

Codomain: Ambient free module of rank 1 over the principal ideal domain ..., Free module morphism defined by the matrix

[0]

[1]

Domain: Ambient free module of rank 2 over the principal ideal domain ...

Codomain: Ambient free module of rank 1 over the principal ideal domain ...)

"""

try:

return self.__basis

except AttributeError:

M = self._matrix_space()

B = M.basis()

self.__basis = tuple([self(x) for x in B])

return self.__basis

def identity(self):

r"""

Return identity morphism in an endomorphism ring.

EXAMPLES::

sage: V=FreeModule(ZZ,5)

sage: H=V.Hom(V)

sage: H.identity()

Free module morphism defined by the matrix

[1 0 0 0 0]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

Domain: Ambient free module of rank 5 over the principal ideal domain ...

Codomain: Ambient free module of rank 5 over the principal ideal domain ...

"""

if self.is_endomorphism_set():

return self(identity_matrix(self.base_ring(),self.domain().rank()))

else:

raise TypeError("Identity map only defined for endomorphisms. Try natural_map() instead.")

Coverage for local/lib/python2.7/site-packages/sage/modules/free_module_homspace.py : 66%

44 statements 29 run 15 missing 0 excluded