True
"""
+ # As with the _element_constructor_() method on the
+ # algebra... even in a subspace of a subspace, the basis
+ # elements belong to the ambient space. As a result, only one
+ # level of coordinate_vector() is needed, regardless of how
+ # deeply we're nested.
W = self.parent().vector_space()
V = self.parent().superalgebra().vector_space()
- A = W.basis_matrix().transpose()
- W_coords = A*self.to_vector()
- V_coords = V.coordinate_vector(W_coords)
+
+ # Multiply on the left because basis_matrix() is row-wise.
+ ambient_coords = self.to_vector()*W.basis_matrix()
+ V_coords = V.coordinate_vector(ambient_coords)
return self.parent().superalgebra().from_vector(V_coords)
except ValueError:
prefix = prefixen[0]
- basis_vectors = [ b.to_vector() for b in basis ]
- superalgebra_basis = [ self._superalgebra.from_vector(b)
- for b in basis_vectors ]
-
# If our superalgebra is a subalgebra of something else, then
# these vectors won't have the right coordinates for
# V.span_of_basis() unless we use V.from_vector() on them.
- W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
+ W = V.span_of_basis( V.from_vector(b.to_vector()) for b in basis )
- n = len(superalgebra_basis)
+ n = len(basis)
mult_table = [[W.zero() for i in range(n)] for j in range(n)]
for i in range(n):
for j in range(n):
- product = superalgebra_basis[i]*superalgebra_basis[j]
+ product = basis[i]*basis[j]
# product.to_vector() might live in a vector subspace
# if our parent algebra is already a subalgebra. We
# use V.from_vector() to make it "the right size" in
product_vector = V.from_vector(product.to_vector())
mult_table[i][j] = W.coordinate_vector(product_vector)
- natural_basis = tuple( b.natural_representation()
- for b in superalgebra_basis )
+ natural_basis = tuple( b.natural_representation() for b in basis )
self._vector_space = W
if elt not in self.superalgebra():
raise ValueError("not an element of this subalgebra")
- # The extra hackery is because foo.to_vector() might not
- # live in foo.parent().vector_space()!
- coords = sum( a*b for (a,b)
- in zip(elt.to_vector(),
- self.superalgebra().vector_space().basis()) )
- return self.from_vector(self.vector_space().coordinate_vector(coords))
+ # The extra hackery is because foo.to_vector() might not live
+ # in foo.parent().vector_space()! Subspaces of subspaces still
+ # have user bases in the ambient space, though, so only one
+ # level of coordinate_vector() is needed. In other words, if V
+ # is itself a subspace, the basis elements for W will be of
+ # the same length as the basis elements for V -- namely
+ # whatever the dimension of the ambient (parent of V?) space is.
+ V = self.superalgebra().vector_space()
+ W = self.vector_space()
+
+ # Multiply on the left because basis_matrix() is row-wise.
+ ambient_coords = elt.to_vector()*V.basis_matrix()
+ W_coords = W.coordinate_vector(ambient_coords)
+ return self.from_vector(W_coords)