# Associated Vector Space

The functions `VectorSpace`, `DualVectorSpace`, and `Lattice` return the underlying vector space, dual vector space, and lattice associated to a space of modular symbols. A space of modular symbols is represented internally as a subspace of a vector space, and a subspace of the linear dual of the vector space. To carry along the subspace of the linear dual is useful in many computations; one example is efficient computation of Hecke operators. When the base field is ${\mathbb{Q}}$, the lattice comes from the natural integral structure on modular symbols.

## `VectorSpace(M): ModSym -> ModTupFld, Map, Map`

The vector space $V$ underlying the space of modular symbols $M$, the map $V\rightarrow M$, and the map $M\rightarrow V$.

## `DualVectorSpace(M): ModSym -> ModTupFld`

The subspace of the linear dual of `VectorSpace(AmbientSpace(M))` that is isomorphic to the space of modular symbols $M$ as a module over the Hecke algebra.

## `Lattice(M): ModSym -> Lat`

The lattice generated by the integral modular symbols in the vector space representation of the space of modular symbols $M$. This is the lattice generated by all modular symbols $X^iY^{k-2-i}\{a,b\}$. The base field of $M$ must be `RationalField()`.

## `Example: Representation (ex-fb3783)`

```magma
> M := ModularSymbols(DirichletGroup(11).1,3); M;
Full modular symbols space of level 11, weight 3, character $.1, and
dimension 4 over Rational Field
> VectorSpace(M);
Full Vector space of degree 4 over Rational Field
Mapping from: Full Vector space of degree 4 over Rational Field to
ModSym: M given by a rule [no inverse]
Mapping from: ModSym: M to Full Vector space of degree 4 over Rational
Field given by a rule [no inverse]
> Basis(VectorSpace(CuspidalSubspace(M)));
[
    ( 0  1  0 -1),
    ( 0  0  1 -1)
]
> Basis(VectorSpace(EisensteinSubspace(M)));
[
    (   1    0 -2/3 -1/3),
    ( 0  1 -5 -2)
]
> Lattice(CuspidalSubspace(M));
Lattice of rank 2 and degree 4
Basis:
( 0  1 -1  0)
( 0  1  1 -2)
Basis Denominator: 2
Mapping from: Lattice of rank 2 and degree 4 to Modular symbols space
of level 11, weight 3, character $.1, and dimension 2 over Rational
Field given by a rule [no inverse]
> Basis(Lattice(EisensteinSubspace(M)));
[
    (   0  1/2 -5/2   -1),
    (   3 -1/2  1/2    0)
]

```