Basic Operations#

Accessing Vector Space Invariants#

V . i: ModTupFld, RngIntElt -> ModTupFldElt#: Given a vector space \(V\) and a positive integer \(i\), return the \(i\)-th generating element of \(V\).

CoefficientField(V): ModTupFld -> Fld#
BaseField(V): ModTupFld -> Fld#: Given a \(K\)-vector space \(V\), return the field \(K\).

Degree(V): ModTupFld -> RngIntElt#: Given a \(K\)-vector space \(V\) which is a subspace of \(K^{(n)}\), return \(n\).

Degree(u): ModTupFldElt -> RngIntElt#: Given a vector \(u\) belonging to a subspace of the vector space \(K^{(n)}\), return \(n\).

Dimension(V): ModTupFld -> RngIntElt#: The dimension of the vector space \(V\).

Generators(V): ModTupFld -> { ModElt}#: The generators for the vector space \(V\), returned as a set.

NumberOfGenerators(M): ModTupFld -> RngIntElt#
Ngens(M): ModTupFld -> RngIntElt#: The number of generators for the vector space \(V\).

OverDimension(V): ModTupFld -> RngIntElt#: Given a \(K\)-vector space \(V\) which is a subspace of \(K^{(n)}\), return \(n\).

OverDimension(u): ModTupFldElt -> RngIntElt#: Given a vector \(u\) belonging to a subspace of the vector space \(K^{(n)}\), return \(n\).

Generic(V): ModFld -> ModFld#: The generic vector space containing \(V\), i.e. the full vector space in which \(V\) is naturally embedded.

Parent(V): ModFld -> SetPow#: The power structure for the vector space \(V\) (the set consisting of all finite dimensional vector spaces).

Membership and Equality#

v in V: ModTupFldElt, ModTupFld -> BoolElt#: Returns true if the element \(v\) lies in the vector space \(V\), where \(v\) and \(V\) belong to a common space.

v notin V: ModTupFldElt, ModTupFld -> BoolElt#: Returns true if the element \(v\) does not lie in the vector space \(V\), where \(v\) and \(V\) belong to a common space.

U subset V: ModTupFld, ModTupFld -> BoolElt#: Returns true if the \(K\)-vector space \(U\) is contained in the \(K\)-vector space \(V\), where \(U\) and \(V\) are subspaces of some common vector space.

U notsubset V: ModTupFld, ModTupFld -> BoolElt#: Returns true if the \(K\)-vector space \(U\) is not contained in the \(K\)-vector space \(V\), where \(U\) and \(V\) are subspaces of some common vector space.

U eq V: ModTupFld, ModTupFld -> BoolElt#: Returns true if the subspaces \(U\) and \(V\) are equal, where \(U\) and \(V\) belong to a common vector space.

U ne V: ModTupFld, ModTupFld -> BoolElt#: Returns true if the subspaces \(U\) and \(V\) are not equal, where \(U\) and \(V\) belong to a common vector space.

Operations on Subspaces#

U + V: ModTupFld, ModTupFld -> ModTupFld#: Sum of the subspaces \(U\) and \(V\), where \(U\) and \(V\) must be subspaces of a common vector space.

U meet V: ModTupFld, ModTupFld -> ModTupFld#: Intersection of the subspaces \(U\) and \(V\), where \(U\) and \(V\) must be subspaces of a common vector space.

U meet:= V: ModTupFld, ModTupFld -> ModTupFld#: Replace \(U\) with the intersection of the subspaces \(U\) and \(V\), where \(U\) and \(V\) must be subspaces of a common vector space.

&meet S: [ ModTupFld ] -> ModTupFld#: Intersection of the subspaces of the set or sequence \(S\), which must be subspaces of a common vector space.

TensorProduct(U, V): ModTupFld, ModTupFld -> FldElt#: The tensor (Kronecker) product of the vector spaces \(U\) and \(V\), generated by all the tensor products of elements of \(U\) by elements of \(V\). The resulting vector space has degree equal to the product of the degrees of \(U\) and \(V\).

Complement(V, U): ModTupFld, ModTupFld -> ModTupFld#

Given a subspace \(U\) of the vector space \(V\), construct a complement for \(U\) in \(V\) (a subspace of \(V\)).

Note that this complement is always with respect to the standard inner product, and thus need not be orthogonal with respect to a given one.

Transversal(V, U): ModTupFld, ModTupFld -> { ModTupFldELt}#: Given a subspace \(U\) of the vector space \(V\) over a finite field, return a transversal for \(U\) in \(V\) as a set of vectors.