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.