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)
ambient_space,
base_field,
base_ring,
basis,
change_ring,
coordinate_vector,
coordinates,
degree,
dimension,
echelonized_basis,
gen,
intersection,
is_ambient,
is_dense,
is_full,
is_sparse,
is_subspace,
linear_combination_of_basis,
matrix,
ngens,
random,
random_element,
subspace,
subspace_with_basis,
vector,
vector_space,
zero_subspace,
zero_vector
Further documentation:
R) |
Change this vector space to be a vector space over R by coercing the basis vectors into R.
v) |
Write v in terms of the user basis for self.
Returns a vector 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.
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.
other) |
Return the intersection of self and other, which must be subspaces of a common ambient space.
sage: V = VectorSpace(RationalField(),3) sage: W1 = V.subspace([V.gen(0), V.gen(0) + V.gen(1)]) sage: W2 = V.subspace([V.gen(1), V.gen(2)]) sage: W1.intersection(W2) Vector space of degree 3, dimension 1 over Rational Field Basis matrix: [0 1 0] sage: W2.intersection(W1) Vector space of degree 3, dimension 1 over Rational Field Basis matrix: [0 1 0] sage: V.intersection(W1) Vector space of degree 3, dimension 2 over Rational Field Basis matrix: [1 0 0] [0 1 0] sage: W1.intersection(V) Vector space of degree 3, dimension 2 over Rational Field Basis matrix: [1 0 0] [0 1 0] sage: Z = V.subspace([]) sage: W1.intersection(Z) Vector space of degree 3, dimension 0 over Rational Field Basis matrix: []
other) |
True if this vector space is a subspace of other.
sage: V = VectorSpace(RationalField(),3) sage: W = V.subspace([V.gen(0), V.gen(0) + V.gen(1)]) sage: W2 = V.subspace([V.gen(1)]) sage: W.is_subspace(V) True sage: W2.is_subspace(V) True sage: W.is_subspace(W2) False sage: W2.is_subspace(W) True
v) |
Return the linear combination of the basis for self obtained from the coordinates of v.
[X=True], [prob=1.0], [coerce=[-2, -1, 1, 2]]) |
Returns a random element of self.
[X=True], [prob=1.0], [coerce=[-2, -1, 1, 2]]) |
Returns a random element of self.
gens, [check=True]) |
Create a subspace of self.
INPUT: gens -- a list of vector in self check -- whether or not to verify that each vector is 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: V = VectorSpace(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]
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: V = VectorSpace(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]
Instances of class VectorSpace_generic also have the following special functions:
__add__,
__call__,
__cmp__,
__contains__
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