**The material described in this chapter is subject to change.**

Vector spaces form another important domain in **GAP3**. They may be given
in any representation whenever the underlying set of elements forms a
vector space in terms of linear algebra. Thus, for example, one may
construct a vector space by defining generating matrices over a field or
by using the base of a field extension as generators. More complex
constructions may fake elements of a vector space by specifying records
with appropriate operations. A special type of vector space, that is
implemented in the **GAP3** library, handles the case where the elements
are lists over a field. This type is the so called `RowSpace`

(see Row
Spaces for details).

General vector spaces are created using the function `VectorSpace`

(see
VectorSpace) and they are represented as records that contain all
necessary information to deal with the vector space. The components
listed in Vector Space Records are common for all vector spaces, but
special types of vector spaces, such as the row spaces, may use
additional entries to store specific data.

The following sections contain descriptions of functions and operations defined for vector spaces.

The next sections describe functions to compute a base (see Base) and the dimension (see Dimension) of a vector space over its field.

The next sections describe how to calculate linear combinations of the elements of a base (see LinearCombination) and how to find the coefficients of an element of a vector space when expressed as a linear combination in the current base (see Coefficients).

The functions described in this chapter are implemented in the file
`LIBNAME/"vecspace.g"`

.

- VectorSpace
- IsVectorSpace
- Vector Space Records
- Set Functions for Vector Spaces
- IsSubspace
- Base
- AddBase
- Dimension
- LinearCombination
- Coefficients

`VectorSpace( `

`generators`, `field` )

Let `generators` be a list of objects generating a vector space over the
field `field`. Then `VectorSpace`

returns this vector space represented
as a **GAP3** record.

gap> f := GF( 3^2 ); GF(3^2) gap> m := [ [ f.one, f.one ], [ f.zero, f.zero ] ]; [ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ] gap> n := [ [ f.one, f.zero ], [ f.zero, f.one ] ]; [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ] gap> VectorSpace( [ m, n ], f ); VectorSpace( [ [ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ], [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ] ], GF(3^2) )

`VectorSpace( `

`generators`, `field`, `zero` )

`VectorSpace`

returns the vector space generated by `generators` over the
field `field` having `zero` as the uniquely determined neutral element.
This call of `VectorSpace`

always is requested if `generators` is the
empty list.

gap> VectorSpace( [], f, [ [ f.zero, f.zero ], [ f.zero, f.zero ] ] ); VectorSpace( [ ], GF(3^2), [ [ 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3) ] ] )

`IsVectorSpace( `

`obj` )

`IsVectorSpace`

returns `true`

if `obj`, which can be an object of
arbitrary type, is a vector space and `false`

otherwise.

A vector space is represented as a **GAP3** record having several entries
to hold some necessary information about the vector space.

Basically a vector space record is constructed using the function
`VectorSpace`

although one may create such a record by hand. Furthermore
vector space records may be returned by functions described here or
somewhere else in this manual.

Once a vector space record is created you are free to add components, but
you should never alter existing entries, especially `generators`

, `field`

and `zero`

.

The following list mentions all components that are requested for a
vector space `V`.

`generators`

:-

a list of elements generating the vector space`V`.

`field`

:-

the field over which the vector space`V`is written.

`zero`

:-

the zero element of the vector space.

`isDomain`

:-

always`true`

, because vector spaces are domains.

`isVectorSpace`

:-

always`true`

, for obvious reasons.

There are as well some optional components for a vector space record.

`base`

:-

a base for`V`, given as a list of elements of`V`.

`dimension`

:-

the dimension of`V`which is the length of a base of`V`.

As mentioned before, vector spaces are domains. So all functions that exist for domains may also be applied to vector spaces. This and the following chapters give further information on the implementation of these functions for vector spaces, as far as they differ in their implementation from the general functions.

`Elements( `

`V` )

The elements of a vector space `V` are computed by producing all linear
combinations of the generators of `V`.

`Size( `

`V` )

The size of a vector space `V` is determined by calculating the dimension
of `V` and looking at the field over which it is written.

`IsFinite( `

`V` )

A vector space in **GAP3** is finite if it contains only its zero element
or if the field over which it is written is finite. This characterisation
is true here, as in **GAP3** all vector spaces have a finite dimension.

`Intersection( `

`V`, `W` )

The intersection of vector spaces is computed by finding a base for the
intersection of the sets of their elements. One may consider the
algorithm for finding a base of a vector space `V` as another way to
write `Intersection( `

.
`V`, `V` )

`IsSubspace( `

`V`, `W` )

`IsSubspace`

tests whether the vector space `W` is a subspace of `V`. It
returns `true`

if `W` lies in `V` and `false`

if it does not.

The answer to the question is obtained by testing whether all the
generators of `W` lie in `V`, so that, for the general case of vector
space handling, a list of all the elements of `V` is constructed.

`Base( `

`V` )

`Base`

computes a base of the given vector space `V`. The result is
returned as a list of elements of the vector space `V`.

The base of a vector space is defined to be a minimal generating set. It
can be shown that for a given vector space `V` each base has the same
number of elements, which is called the dimension of `V` (see
Dimension).

Unfortunately, no better algorithm is known to compute a base in general than to browse through the list of all elements of the vector space. So be careful when using this command on plain vector spaces.

gap> f := GF(3); GF(3) gap> m1 := [[ f.one, f.one, f.zero, f.zero ]]; [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ] gap> m2 := [[ f.one, f.one, f.one, f.zero ]]; [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ] gap> V := VectorSpace( [ m1, m2, m1+m2 ], GF(3) ); VectorSpace( [ [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ], [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ], [ [ Z(3), Z(3), Z(3)^0, 0*Z(3) ] ] ], GF(3) ) gap> Base( V ); [ [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ], [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ] ] gap> Dimension( V ); 2

`AddBase( `

`V`, `base` )

`AddBase`

attaches a user-supplied base for the vector space `V` to the
record that represents `V`.

Most of the functions for vector spaces make use of a base (see
LinearCombination, Coefficients). These functions get access to a
base using the function `Base`

, which normally computes a base for the
vector space using an appropriate algorithm. Once a base is computed it
will always be reused, no matter whether there is a more interesting base
available or not.

`AddBase`

installs a given `base` for `V` by overwriting any other base
of the vector space that has been installed before. So after `AddBase`

has successfully been used, `base` will be used whenever `Base`

is called
with `V` as argument.

Calling `AddBase`

with a `base` which is not a base for `V` might produce
unpredictable results in following computations.

gap> f := GF(3); GF(3) gap> m1 := [[ f.one, f.one, f.zero, f.zero ]]; [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ] gap> m2 := [[ f.one, f.one, f.one, f.zero ]]; [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ] gap> V := VectorSpace( [ m1, m2, m1+m2 ], GF(3) ); VectorSpace( [ [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ], [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ], [ [ Z(3), Z(3), Z(3)^0, 0*Z(3) ] ] ], GF(3) ) gap> Base( V ); [ [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ], [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ] ] gap> AddBase( V, [ m1, m1+m2 ] ); gap> Base( V ); [ [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ], [ [ Z(3), Z(3), Z(3)^0, 0*Z(3) ] ] ]

`Dimension( `

`V` )

`Dimension`

computes the dimension of the given vector space `V` over its
field.

The dimension of a vector space `V` is defined to be the length of a
minimal generating set of `V`, which is called a base of `V` (see
Base).

The implementation of `Dimension`

strictly follows its above definition,
so that this function will always determine a base of `V`.

gap> f := GF( 3^4 ); GF(3^4) gap> f.base; [ Z(3)^0, Z(3^4), Z(3^4)^2, Z(3^4)^3 ] gap> V := VectorSpace( f.base, GF( 3 ) ); VectorSpace( [ Z(3)^0, Z(3^4), Z(3^4)^2, Z(3^4)^3 ], GF(3) ) gap> Dimension( V ); 4

`LinearCombination( `

`V`, `cf` )

`LinearCombination`

computes the linear combination of the base elements
of the vector space `V` with coefficients `cf`.

`cf` has to be a list of elements of `V`.field, the field over which the
vector space is written. Its length must be equal to the dimension of `V`
to make sure that one coefficient is specified for each element of the
base.

`LinearCombination`

will use that base of `V` which is returned when
applying the function `Base`

to `V` (see Base). To perform linear
combinations of different bases use `AddBase`

to specify which base
should be used (see AddBase).

gap> f := GF( 3^4 ); GF(3^4) gap> f.base; [ Z(3)^0, Z(3^4), Z(3^4)^2, Z(3^4)^3 ] gap> V := VectorSpace( f.base, GF( 3 ) ); VectorSpace( [ Z(3)^0, Z(3^4), Z(3^4)^2, Z(3^4)^3 ], GF(3) ) gap> LinearCombination( V, [ Z(3), Z(3)^0, Z(3), 0*Z(3) ] ); Z(3^4)^16 gap> Coefficients( V, f.root ^ 16 ); [ Z(3), Z(3)^0, Z(3), 0*Z(3) ]

`Coefficients( `

`V`, `v` )

`Coefficients`

computes the coefficients that have to be used to write
`v` as a linear combination in the base of `V`.

To make sure that this function produces the correct result, `v` has to
be an element of `V`. If `v` does not lie in `V` the result is
unpredictable.

The result of `Coefficients`

is returned as a list of elements of the
field over which the vector space `V` is written. Of course, the length
of this list equals the dimension of `V`.

gap> f := GF( 3^4 ); GF(3^4) gap> f.base; [ Z(3)^0, Z(3^4), Z(3^4)^2, Z(3^4)^3 ] gap> V := VectorSpace( f.base, GF( 3 ) ); VectorSpace( [ Z(3)^0, Z(3^4), Z(3^4)^2, Z(3^4)^3 ], GF(3) ) gap> Dimension( V ); 4 gap> Coefficients( V, f.root ^ 16 ); [ Z(3), Z(3)^0, Z(3), 0*Z(3) ]

gap3-jm

24 Apr 2021