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)
4
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 ...

>>> from sage.all import *
>>> M = FreeModule(IntegerRing(), Integer(2))
>>> E = End(M)
>>> B = E.basis()
>>> len(B)
4
>>> B[Integer(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 noetherian rings
       and infinite enumerated sets and metric spaces)
sage: B = H.basis()
sage: len(B)
6
sage: B[0]
Free module morphism defined by the matrix
[1 0]
[0 0]
[0 0]...
[Python] 
>>> from sage.all import *
>>> V3 = FreeModule(IntegerRing(), Integer(3))
>>> V2 = FreeModule(IntegerRing(), Integer(2))
>>> H = Hom(V3, V2)
>>> 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 noetherian rings
       and infinite enumerated sets and metric spaces)
>>> B = H.basis()
>>> len(B)
6
>>> B[Integer(0)]
Free module morphism defined by the matrix
[1 0]
[0 0]
[0 0]...
class sage.modules.free_module_homspace.FreeModuleHomspace(X, Y, category=None, check=True, base=None)[source]

Bases: HomsetWithBase

basis(side='left')[source]

Return a basis for this space of free module homomorphisms.

INPUT:

  • side – side of the vectors acted on by the matrix (default: 'left')

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 ...)
sage: H.basis("right")
(Free module morphism defined as left-multiplication 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 as left-multiplication 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 ...)

>>> from sage.all import *
>>> H = Hom(ZZ**Integer(2), ZZ**Integer(1))
>>> 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 ...)
>>> H.basis("right")
(Free module morphism defined as left-multiplication 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 as left-multiplication 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 ...)
identity(side='left')[source]

Return identity morphism in an endomorphism ring.

INPUT:

  • side – side of the vectors acted on by the matrix (default: 'left')

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 ...

>>> from sage.all import *
>>> V = FreeModule(ZZ,Integer(5))
>>> H = V.Hom(V)
>>> 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 ...
zero(side='left')[source]

INPUT:

  • side – side of the vectors acted on by the matrix (default: 'left')

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]
[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
sage: H.zero("right")
Free module morphism defined as left-multiplication by the matrix
[0 0]
[0 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

>>> from sage.all import *
>>> E = ZZ**Integer(2)
>>> F = ZZ**Integer(3)
>>> H = Hom(E, F)
>>> f = H.zero()
>>> f
Free module morphism defined by the matrix
[0 0 0]
[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
>>> f(E.an_element())
(0, 0, 0)
>>> f(E.an_element()) == F.zero()
True
>>> H.zero("right")
Free module morphism defined as left-multiplication by the matrix
[0 0]
[0 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