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Subspaces, Quotient Spaces and Homomorphisms

Construction of Subspaces

The conventions defining the presentations of subspaces and quotient spaces are as follows:

If V has been created using the function VectorSpace or MatrixSpace, then every subspace and quotient space of V is given in terms of a basis consisting of elements of V, i.e. by means of an embedded basis.

If V has been created using the function RModule, then every subspace and quotient space of V is given in terms of a reduced basis.

sub<V | L> : ModTupFld, List -> ModTupFld

Given a K-vector space V, construct the subspace U generated by the elements of V specified by the list L. Each term L_i of the list L must be an expression defining an object of one of the following types:

(a)
A sequence of n elements of K defining an element of V;
(b)
A set or sequence whose terms are elements of V;
(c)
A subspace of V;
(d)
A set or sequence whose terms are subspaces of V.

The generators stored for U consist of the vectors specified by terms L_i together with the stored generators for subspaces specified by terms of L_i. Repetitions of a vector and occurrences of the zero vector are removed (unless U is the trivial subspace).
The constructor returns the subspace U and the inclusion homomorphism f : U -> V. If V is of embedded type, the basis constructed for U consists of elements of V. If V is of standard type, a standard basis is constructed for U.

Morphism(U, V) : ModTupFld, ModTupFld -> RModMatElt

Assuming the vector space U has been created as a subspace of V, the function returns the matrix defining the embedding of U into V.

Example `ModFld_Subspace1 (H29E8)`

The ternary Golay code is a six-dimensional subspace of the vector space K⁽¹¹⁾, where K is GF(3). This subspace is first constructed in the space constructed by the VectorSpace function.

> K11 := VectorSpace(FiniteField(3), 11);
> G3 := sub< K11  |
>     [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] >;
> G3;
Vector space of degree 11, dimension 6 over GF(3)
Generators:
(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)
Echelonized basis:
(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)

Example `ModFld_Subspace2 (H29E9)`

We now construct the ternary Golay code starting with the vector space constructed using the RModule function. In this case the subspace is presented on a reduced basis.

> K11 := RModule(FiniteField(3), 11);
> G3 := sub< K11 |
>     [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] >;
> G3;
KModule G3 of dimension 6 with base ring GF(3)
> Basis(G3);
[
    G3: (1 0 0 0 0 0),
    G3: (0 1 0 0 0 0),
    G3: (0 0 1 0 0 0),
    G3: (0 0 0 1 0 0),
    G3: (0 0 0 0 1 0),
    G3: (0 0 0 0 0 1)
]
> f := Morphism(G3, K11);
> f;
[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]

Construction of Quotient Vector Spaces

quo<V | L> : ModTupFld, List -> ModTupFld, Map

Given a K-vector space V, construct the quotient vector space W = V/U, where U is the subspace generated by the elements of V specified by the list L. Each term L_i of the list L must be an expression defining an object of one of the following types:

(a)
A sequence of n elements of K defining an element of V;
(b)
A set or sequence whose terms are elements of V;
(c)
A subspace of V;
(d)
A set or sequence whose terms are subspaces of V.

The generators constructed for U consist of the elements specified by terms L_i together with the stored generators for subspaces specified by terms of L_i.
The constructor returns the quotient space W and the natural homomorphism f : V -> W.

V / U : ModTupFld, ModTupFld -> ModTupFld, Map

Given a subspace U of the vector space V, construct the quotient space W of V by U. If r is defined to be dim(V) - dim(U), then W is created as an r-dimensional vector space relative to the standard basis.
The constructor returns the quotient space W and the natural homomorphism f : V -> W.

Example `ModFld_Quotients1 (H29E10)`

We construct the quotient of K⁽¹¹⁾ by the Golay code.

> K11 := VectorSpace(FiniteField(3), 11);
> Q3, f := quo< K11 |
>  [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] >;
> Q3;
Full Vector space of degree 5 over GF(3)
> f;
Mapping from: ModTupFld: K11 to ModTupFld: Q3

Example `ModFld_Quotients2 (H29E11)`

If we wished to construct this quotient of K⁽¹¹⁾ as a subspace of the original space, we could do so using the Complement function.

> K11 := VectorSpace(FiniteField(3), 11);
> S := sub< K11 |
>     [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] >;
> Complement(K11, S);
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)

Example `ModFld_Quotients3 (H29E12)`

We construct a subspace and its quotient space in Q^{(3 x 4)}.

> Q := RationalField();
> Q3 := VectorSpace(Q, 3);
> Q4 := VectorSpace(Q, 4);
> H34 := Hom(Q3, Q4);
> a := H34 ! [ 2, 0, 1, -1/2,  1, 0, 3/2, 4,  4/5, 6/7, 0, -1/3];
> b := H34 ! [ 1/2, -3, 0, 5,  1/3, 2, 4/5, 0,  5, -1, 5, 7];
> c := H34 ! [ -1, 4/9, 1, -4,  5, -5/6, -3/2, 0,  4/3, 7, 0, 7/9];
> d := H34 ! [ -3, 5, 1/3, -1/2,  2/3, 4, -2, 0,  0, 4, -1, 0];
> a, b, c, d;
[   2    0    1 -1/2]
[   1    0  3/2    4]
[ 4/5  6/7    0 -1/3]
[1/2  -3   0   5]
[1/3   2 4/5   0]
[  5  -1   5   7]
[  -1  4/9    1   -4]
[   5 -5/6 -3/2    0]
[ 4/3    7    0  7/9]
[  -3    5  1/3 -1/2]
[ 2/3    4   -2    0]
[   0    4   -1    0]
> U := sub< H34 | a, b, c, d >;
> U:Maximal;
KMatrixSpace of 3 by 4 GHom matrices and dimension 4 over Rational Field
Echelonized basis:
[1    0    0    0]
[-33872/30351    -5164/10117    42559/50585    11560/10117]
[-10514/10117    -121582/70819    -8476/10117    -48292/30351]
[           0            1            0            0]
[ -7797/10117   4803/10117 12861/101170   5940/10117]
[ -7818/10117 -38214/70819  -7821/10117 -10967/10117]
[           0            0            1            0]
[ 31261/10117  28101/20234  -2157/20234  18552/10117]
[161802/50585 291399/70819  20088/10117  33419/10117]
[          0           0           0           1]
[-8624/30351  7445/10117  7696/50585  2408/10117]
[32388/50585 -3562/10117  6272/10117 27580/30351]
> W := H34/U;
> W;
Full Vector space of degree 8 over Rational Field

V2.28, 13 July 2023

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