EXAMPLES: We create 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 ...
We create and
compute a basis.
sage: V3 = VectorSpace(RationalField(),3)
sage: V2 = VectorSpace(RationalField(),2)
sage: H = Hom(V3,V2)
sage: H
Set of Morphisms from Vector space of dimension 3 over Rational Field
to Vector space of dimension 2 over Rational Field in Category of
vector spaces over Rational Field
sage: B = H.basis()
sage: len(B)
6
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 = (QQ^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]...
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.
Return underlying matrix space that contains the matrices that define the homomorphisms in this free module homspace.
EXAMPLES:
sage: H = Hom(QQ^3, QQ^2)
sage: H._matrix_space()
Full MatrixSpace of 3 by 2 dense matrices over Rational Field
Return a basis for this space of free module homomorphisms.
EXAMPLES:
sage: H = Hom(QQ^2, QQ^1)
sage: H.basis()
(Free module morphism defined by the matrix
[1]
[0]
Domain: Vector space of dimension 2 over Rational Field
Codomain: Vector space of dimension 1 over Rational Field,
Free module morphism defined by the matrix
[0]
[1]
Domain: Vector space of dimension 2 over Rational Field
Codomain: Vector space of dimension 1 over Rational Field)
Return identity morphism in an endomorphism ring.
EXAMPLE:
sage: V=VectorSpace(QQ,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: Vector space of dimension 5 over Rational Field
Codomain: Vector space of dimension 5 over Rational Field
Return True if x is a Free module homspace.
EXAMPLES:
sage: H = Hom(QQ^3, QQ^2)
sage: sage.modules.free_module_homspace.is_FreeModuleHomspace(H)
True
sage: sage.modules.free_module_homspace.is_FreeModuleHomspace(2)
False