def superset_basis(C, T):
'''
Input:
- C: linearly independent set of Vecs
- T: set of Vecs such that every Vec in C is in Span(T)
Output:
Linearly independent set S consisting of all Vecs in C and some in T
such that the span of S is the span of T (i.e. S is a basis for the span
of T).
Example:
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = superset_basis({a0, a3}, {a0, a1, a2})
>>> a0 in sb and a3 in sb
True
>>> is_independent(sb)
True
>>> all(x in [a0,a1,a2,a3] for x in sb)
True
'''
S = {x for x in C}
for x in T:
S.add(x)
if is_superfluous(S,x):
S.remove(x)
return S
def superset_basis(C, T):
'''
Input:
- C: linearly independent set of Vecs
- T: set of Vecs such that every Vec in S is in Span(T)
Output:
Linearly independent set S consisting of all Vecs in C and some in T
such that the span of S is the span of T (i.e. S is a basis for the span
of T).
Example:
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = superset_basis({a0, a3}, {a0, a1, a2})
>>> a0 in sb and a3 in sb
True
>>> is_independent(sb)
True
>>> all(x in [a0,a1,a2,a3] for x in sb)
True
'''
S = {s for s in C}
for t in T:
S.update({t})
if is_superfluous(S,t):
S = S - {t}
return S
def superset_basis(C, T):
'''
Input:
- C: linearly independent set of Vecs
- T: set of Vecs such that every Vec in C is in Span(T)
Output:
Linearly independent set S consisting of all Vecs in C and some in T
such that the span of S is the span of T (i.e. S is a basis for the span
of T).
Example:
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = superset_basis({a0, a3}, {a0, a1, a2})
>>> a0 in sb and a3 in sb
True
>>> is_independent(sb)
True
>>> all(x in [a0,a1,a2,a3] for x in sb)
True
'''
res = C | T
A = res - C
for v in A:
if is_superfluous(res, v):
res.remove(v)
return res
def subset_basis(T):
'''
Input:
- T: a set of Vecs
Output:
- set S containing Vecs from T that is a basis for Span T.
Examples:
The following tests use the procedure is_independent, provided in module independence
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = subset_basis({a0, a1, a2, a3})
>>> len(sb)
3
>>> all(v in [a0, a1, a2, a3] for v in sb)
True
>>> is_independent(sb)
True
>>> b0 = Vec({0,1,2,3},{0:2,1:2,3:4})
>>> b1 = Vec({0,1,2,3},{0:1,1:1})
>>> b2 = Vec({0,1,2,3},{2:3,3:4})
>>> b3 = Vec({0,1,2,3},{3:3})
>>> sb = subset_basis({b0, b1, b2, b3})
>>> len(sb)
3
>>> all(v in [b0, b1, b2, b3] for v in sb)
True
>>> is_independent(sb)
True
>>> D = {'a','b','c','d'}
>>> c0, c1, c2, c3, c4 = Vec(D,{'d': one, 'c': one}), Vec(D,{'d': one, 'a': one, 'c': one, 'b': one}), Vec(D,{'a': one}), Vec(D,{}), Vec(D,{'d': one, 'a': one, 'b': one})
>>> subset_basis({c0,c1,c2,c3,c4}) == {c0,c1,c2,c4}
True
'''
S = set()
for x in T:
S.add(x)
if is_superfluous(S,x):
S.remove(x)
return S
def subset_basis(T):
'''
Input:
- T: a set of Vecs
Output:
- set S containing Vecs from T that is a basis for Span T.
Examples:
The following tests use the procedure is_independent, provided in module independence
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = subset_basis({a0, a1, a2, a3})
>>> len(sb)
3
>>> all(v in [a0, a1, a2, a3] for v in sb)
True
>>> is_independent(sb)
True
>>> b0 = Vec({0,1,2,3},{0:2,1:2,3:4})
>>> b1 = Vec({0,1,2,3},{0:1,1:1})
>>> b2 = Vec({0,1,2,3},{2:3,3:4})
>>> b3 = Vec({0,1,2,3},{3:3})
>>> sb = subset_basis({b0, b1, b2, b3})
>>> len(sb)
3
>>> all(v in [b0, b1, b2, b3] for v in sb)
True
>>> is_independent(sb)
True
>>> D = {'a','b','c','d'}
>>> c0, c1, c2, c3, c4 = Vec(D,{'d': one, 'c': one}), Vec(D,{'d': one, 'a': one, 'c': one, 'b': one}), Vec(D,{'a': one}), Vec(D,{}), Vec(D,{'d': one, 'a': one, 'b': one})
>>> subset_basis({c0,c1,c2,c3,c4}) == {c0,c1,c2,c4}
True
'''
S = {s for s in T}
for t in T:
if is_superfluous(S,t):
S = S - {t}
return S
