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 x 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 v1, ..., vr, return a sequence [a1, ..., ar] of elements of K giving the coordinates of v relative to the V-basis: v = a1 * v1 + ... + ar * vr.
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, ... 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 ModFld_Basis (H29E13)

These operations will be illustrated in the context of the subspace G3 of the 11-dimensional vector space over GF(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)
V2.28, 13 July 2023