5.1.1.3 VectorSpace_generic Objects

class VectorSpace_generic
The VectorSpace_generic class derives from VectorSpace, and defines functionality for generic vector spaces over an arbitrary base field. (One never instantiates objects of this class.)
VectorSpace_generic( base_field, degree, [sparse=False])

Create the space of all vectors of given degree over base_field.

INPUT:
    base_field -- a field
    degree -- int >= 0, the degree of the vector space 
              (number of components of vectors).
    sparse -- whether or not matrices are given a sparse
representation 
              (default to False)

Instances of class VectorSpace_generic have the following methods (in addition to inherited methods and special methods):

ambient_space,$  $ base_field,$  $ base_ring,$  $ basis,$  $ coordinates,$  $ degree,$  $ dimension,$  $ echelonized_basis,$  $ gen,$  $ is_ambient,$  $ is_dense,$  $ is_full,$  $ is_sparse,$  $ matrix,$  $ ngens,$  $ random,$  $ random_element,$  $ subspace,$  $ subspace_with_basis,$  $ vector,$  $ vector_space,$  $ zero_vector

These methods are defined as follows:

ambient_space( )

base_field( )

base_ring( )

basis( )

coordinates( v)

Write v in terms of the basis for self.

Returns a list c such that if B is the basis for self, then sum c[i] B[i] = v If v is not in self, raises an ArithmeticError exception.

degree( )

dimension( )

echelonized_basis( )

gen( n)

is_ambient( )

is_dense( )

is_full( )

is_sparse( )

matrix( )

ngens( )

random( [X=True], [prob=1.0], [coerce=[-2, -1, 1, 2]])

Returns a random element of self.

random_element( [X=True], [prob=1.0], [coerce=[-2, -1, 1, 2]])

Returns a random element of self.

subspace( gens)

Create a subspace of self.

INPUT:
    gens -- a list of vector in self
OUTPUT:
    VectorSpace -- the subspace spanned by the vectors in the
list gens.
    The basis for the subspace is always put in reduced row
echelon form.

sage: import sage.rings.rings as rings
sage: V = VectorSpace(rings.RationalField(), 3)
sage: B = V.basis()
sage: W = V.subspace([B[0]+B[1], 2*B[1]-B[2]])
sage: W
Vector space of degree 3, dimension 2 over Rational Field
Basis matrix:
[   1    0  1/2]
[   0    1 -1/2]

subspace_with_basis( basis)

Create a subspace of self with given basis.

INPUT:
    basis -- a list of linearly independent vectors

OUTPUT:
    VectorSpace_subspace_with_basis -- the subspace with given
basis.
    The basis for the subspace is always put in reduced row
echelon form.

sage: import sage.rings.rings as rings
sage: V = VectorSpace(rings.RationalField(), 3)
sage: B = V.basis()
sage: W = V.subspace_with_basis([B[0]+B[1], 2*B[1]-B[2]])
sage: W
Vector space of degree 3, dimension 2 over Rational Field
User basis matrix:
[ 1  1  0]
[ 0  2 -1]

vector( [x=True], [coerce_entries=True], [copy=True], [check_element=0])

vector_space( degree)

zero_vector( )

Instances of class VectorSpace_generic also have the following special methods:

__add__( other)

__call__( [entries=True], [coerce_entries=True], [copy=True], [check_element=0])

__cmp__( right)

__contains__( x)

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