Bases

This section is concerned with the construction of bases for vector spaces.

VectorSpaceWithBasis(Q): [ModTupFldElt] -> ModTupFld

VectorSpaceWithBasis(a): AlgMatElt -> ModTupFld

VectorSpaceWithBasis(a): ModMatFldElt -> ModTupFld

KSpaceWithBasis(Q): [ModTupFldElt] -> ModTupFld

KSpaceWithBasis(a): AlgMatElt -> ModTupFld

KSpaceWithBasis(a): ModMatFldElt -> ModTupFld

KModuleWithBasis(Q): [ModTupFldElt] -> ModTupFld

Create a vector space having as basis the terms of \(B\) (rows of \(a\)).

Basis(V): ModTupFld -> [ModTupFldElt]

The current basis for the vector space \(V\), returned as a sequence of vectors.

BasisElement(V, i): ModTupFld, RngIntElt -> ModTupFldElt

The \(i\)-th basis element for the vector space \(V\).

BasisMatrix(V): ModTupFld -> ModMatElt

The current basis for the vector space \(V\), returned as the rows of a matrix belonging to the matrix space \(K^{(m \times n)}\), where \(m\) is the dimension of \(V\) and \(n\) is the over-dimension of \(V\).

Coordinates(V, v): ModTupFld, ModTupFldElt -> [FldElt]

Given a vector \(v\) belonging to the \(r\)-dimensional \(K\)-vector space \(V\), with basis \(v_1, \ldots, v_r\), return a sequence \([a_1, \ldots, a_r]\) of elements of \(K\) giving the coordinates of \(v\) relative to the \(V\)-basis: \(v = a_1*v_1 + \cdots +a_r*v_r\).

Dimension(V): ModTupFld -> RngIntElt

The dimension of the vector space \(V\).

ExtendBasis(Q, U): [ModTupFldElt], ModTupFld -> [ModTupFldElt]

Given a sequence \(Q\) containing \(r\) linearly independent vectors belonging to the vector space \(U\), extend the vectors of \(Q\) to a basis for \(U\). The basis is returned in the form of a sequence \(T\) such that \(T[i] = Q[i], i = 1, \ldots r\).

ExtendBasis(U, V): ModTupFld, ModTupFld -> [ModTupFldElt]

Given an \(r\)-dimensional subspace \(U\) of the vector space \(V\), return a basis for \(V\) in the form of a sequence \(T\) of elements such that the first \(r\) elements correspond to the given basis vectors for \(U\).

IsIndependent(S): { ModTupFldElt} -> BoolElt

Given a set \(S\) of elements belonging to the vector space \(V\), return true if the elements of \(S\) are linearly independent.

IsIndependent(Q): [ ModTupFldElt ] -> BoolElt

Given a sequence \(Q\) of elements belonging to the vector space \(V\), return true if the terms of \(Q\) are linearly independent.

Example: Basis (ex-a4fb0c)

These operations will be illustrated in the context of the subspace \(G3\) of the \(11\)-dimensional vector space over \({\bf F}_{3}\) defining the ternary Golay code.

> V11 := VectorSpace(FiniteField(3), 11);
> G3  := sub< V11 |  [1,0,0,0,0,0,1,1,1,1,1], [0,1,0,0,0,0,0,1,2,2,1],
>                    [0,0,1,0,0,0,1,0,1,2,2], [0,0,0,1,0,0,2,1,0,1,2],
>                    [0,0,0,0,1,0,2,2,1,0,1], [0,0,0,0,0,1,1,2,2,1,0] >;
> Dimension(G3);
    6
> Basis(G3);
[
    (1 0 0 0 0 0 1 1 1 1 1),
    (0 1 0 0 0 0 0 1 2 2 1),
    (0 0 1 0 0 0 1 0 1 2 2),
    (0 0 0 1 0 0 2 1 0 1 2),
    (0 0 0 0 1 0 2 2 1 0 1),
    (0 0 0 0 0 1 1 2 2 1 0)
]
> S := ExtendBasis(G3, V11);
> S;
[
    (1 0 0 0 0 0 1 1 1 1 1),
    (0 1 0 0 0 0 0 1 2 2 1),
    (0 0 1 0 0 0 1 0 1 2 2),
    (0 0 0 1 0 0 2 1 0 1 2),
    (0 0 0 0 1 0 2 2 1 0 1),
    (0 0 0 0 0 1 1 2 2 1 0),
    (0 0 0 0 0 0 1 0 0 0 0),
    (0 0 0 0 0 0 0 1 0 0 0),
    (0 0 0 0 0 0 0 0 1 0 0),
    (0 0 0 0 0 0 0 0 0 1 0),
    (0 0 0 0 0 0 0 0 0 0 1)
]
> C3:= Complement(V11, G3);
> C3;
Vector space of degree 11, dimension 5 over GF(3)
Echelonized basis:
(0 0 0 0 0 0 1 0 0 0 0)
(0 0 0 0 0 0 0 1 0 0 0)
(0 0 0 0 0 0 0 0 1 0 0)
(0 0 0 0 0 0 0 0 0 1 0)
(0 0 0 0 0 0 0 0 0 0 1)
> G3 + C3;
Full Vector space of degree 11 over GF(3)
> G3 meet C3;
Vector space of degree 11, dimension 0 over GF(3)
> x := Random(G3);
> x;
(1 1 2 0 0 1 1 1 1 2 0)
> c := Coordinates(G3, x);
> c;
[ 1, 1, 2, 0, 0, 1 ]
> G3 ! &+[ c[i] * G3.i : i in [1 .. Dimension(G3)]];
(1 1 2 0 0 1 1 1 1 2 0)

Run in calculatorCopy input