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( 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:
field:
zero:
isDomain:true, because vector spaces are domains.
isVectorSpace:true, for obvious reasons.
There are as well some optional components for a vector space record.
base:
dimension:9.4 Set Functions for Vector Spaces
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