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Subspaces, Quotient Spaces and Homomorphisms

Subspaces, Quotient Spaces and Homomorphisms

Subsections

Construction of Subspaces

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

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 -> Map
If the vector space U has been created as a subspace of V, the function returns the inclusion homomorphism of U into V. If the vector space V has been created as a quotient space of V, the function returns the natural homomorphism of U onto V. Thus, in either case, Morphism returns the second value returned by the sub and quo constructors. The homomorphism is returned as a matrix.

Example KMod_Subspace1 (H41E7)

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] >;
>print 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 KMod_Subspace2 (H41E8)

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
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
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(M) - dim(N), then W is created as an r-dimensional vector space relative to the standard basis.

Example KMod_Quotients1 (H41E9)

We construct the quotient of K^((11)) by the 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 KMod_Quotients2 (H41E10)

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 KMod_Quotients3 (H41E11)

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


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