Quotients of finite rank free modules over a field.¶

class
sage.modules.quotient_module.
FreeModule_ambient_field_quotient
(domain, sub, quotient_matrix, lift_matrix, inner_product_matrix=None)¶ Bases:
sage.modules.free_module.FreeModule_ambient_field
A quotient \(V/W\) of two vector spaces as a vector space.
To obtain \(V\) or \(W\) use
self.V()
andself.W()
.EXAMPLES:
sage: k.<i> = QuadraticField(1) sage: A = k^3; V = A.span([[1,0,i], [2,i,0]]) sage: W = A.span([[3,i,i]]) sage: U = V/W; U Vector space quotient V/W of dimension 1 over Number Field in i with defining polynomial x^2 + 1 where V: Vector space of degree 3 and dimension 2 over Number Field in i with defining polynomial x^2 + 1 Basis matrix: [ 1 0 i] [ 0 1 2] W: Vector space of degree 3 and dimension 1 over Number Field in i with defining polynomial x^2 + 1 Basis matrix: [ 1 1/3*i 1/3*i] sage: U.V() Vector space of degree 3 and dimension 2 over Number Field in i with defining polynomial x^2 + 1 Basis matrix: [ 1 0 i] [ 0 1 2] sage: U.W() Vector space of degree 3 and dimension 1 over Number Field in i with defining polynomial x^2 + 1 Basis matrix: [ 1 1/3*i 1/3*i] sage: U.quotient_map() Vector space morphism represented by the matrix: [ 1] [3*i] Domain: Vector space of degree 3 and dimension 2 over Number Field in i with defining polynomial x^2 + 1 Basis matrix: [ 1 0 i] [ 0 1 2] Codomain: Vector space quotient V/W of dimension 1 over Number Field in i with defining polynomial x^2 + 1 where V: Vector space of degree 3 and dimension 2 over Number Field in i with defining polynomial x^2 + 1 Basis matrix: [ 1 0 i] [ 0 1 2] W: Vector space of degree 3 and dimension 1 over Number Field in i with defining polynomial x^2 + 1 Basis matrix: [ 1 1/3*i 1/3*i] sage: Z = V.quotient(W) sage: Z == U True
We create three quotient spaces and compare them:
sage: A = QQ^2 sage: V = A.span_of_basis([[1,0], [1,1]]) sage: W0 = V.span([V.1, V.0]) sage: W1 = V.span([V.1]) sage: W2 = V.span([V.1]) sage: Q0 = V/W0 sage: Q1 = V/W1 sage: Q2 = V/W2 sage: Q0 == Q1 False sage: Q1 == Q2 True

V
()¶ Given this quotient space \(Q = V/W\), return \(V\).
EXAMPLES:
sage: M = QQ^10 / [list(range(10)), list(range(2,12))] sage: M.V() Vector space of dimension 10 over Rational Field

W
()¶ Given this quotient space \(Q = V/W\), return \(W\).
EXAMPLES:
sage: M = QQ^10 / [list(range(10)), list(range(2,12))] sage: M.W() Vector space of degree 10 and dimension 2 over Rational Field Basis matrix: [ 1 0 1 2 3 4 5 6 7 8] [ 0 1 2 3 4 5 6 7 8 9]

cover
()¶ Given this quotient space \(Q = V/W\), return \(V\).
This is the same as
V()
.EXAMPLES:
sage: M = QQ^10 / [list(range(10)), list(range(2,12))] sage: M.cover() Vector space of dimension 10 over Rational Field

lift
(x)¶ Lift element of this quotient \(V / W\) to \(V\) by applying the fixed lift homomorphism.
The lift is a fixed homomorphism.
EXAMPLES:
sage: M = QQ^3 / [[1,2,3]] sage: M.lift(M.0) (1, 0, 0) sage: M.lift(M.1) (0, 1, 0) sage: M.lift(M.0  2*M.1) (1, 2, 0)

lift_map
()¶ Given this quotient space \(Q = V / W\), return a fixed choice of linear homomorphism (a section) from \(Q\) to \(V\).
EXAMPLES:
sage: M = QQ^3 / [[1,2,3]] sage: M.lift_map() Vector space morphism represented by the matrix: [1 0 0] [0 1 0] Domain: Vector space quotient V/W of dimension 2 over Rational Field where V: Vector space of dimension 3 over Rational Field W: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3] Codomain: Vector space of dimension 3 over Rational Field

quotient_map
()¶ Given this quotient space \(Q = V / W\), return the natural quotient map from \(V\) to \(Q\).
EXAMPLES:
sage: M = QQ^3 / [[1,2,3]] sage: M.quotient_map() Vector space morphism represented by the matrix: [ 1 0] [ 0 1] [1/3 2/3] Domain: Vector space of dimension 3 over Rational Field Codomain: Vector space quotient V/W of dimension 2 over Rational Field where V: Vector space of dimension 3 over Rational Field W: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 3] sage: M.quotient_map()( (QQ^3)([1,2,3]) ) (0, 0)

relations
()¶ Given this quotient space \(Q = V/W\), return \(W\).
This is the same as
W()
.EXAMPLES:
sage: M = QQ^10 / [list(range(10)), list(range(2,12))] sage: M.relations() Vector space of degree 10 and dimension 2 over Rational Field Basis matrix: [ 1 0 1 2 3 4 5 6 7 8] [ 0 1 2 3 4 5 6 7 8 9]
