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 \(f : U \rightarrow V\). 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: Subspace1 (ex-466045)#

The ternary Golay code is a six-dimensional subspace of the vector space \(K^{(11)}\), where \(K\) is \({\bf F}_{3}\). 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)

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Example: Subspace2 (ex-1f57d1)#

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]

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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 \(f : V \rightarrow W\).

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 \(f : V \rightarrow W\).

Example: Quotients1 (ex-4d668d)#

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

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Example: Quotients2 (ex-e0a05b)#

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)

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Example: Quotients3 (ex-aceeb2)#

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

> 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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