def findbasis(V): while not hw4.is_independent(V): for i in range(len(V)): if hw4.is_superfluous(V,i): V.remove(V[i]) break return V
def findbasis(V):
while not hw4.is_independent(V):
for i in range(len(V)):
if hw4.is_superfluous(V, i):
V.remove(V[i])
break
return V
def morph(S, B):
'''
Input:
- S: a list of distinct Vec instances
- B: a list of linearly independent Vec instances
- Span S == Span B
Output: a list of pairs of vectors to inject and eject
Example:
>>> #This is how our morph works. Yours may yield different results.
>>> S = [list2vec(v) for v in [[1,0,0],[0,1,0],[0,0,1]]]
>>> B = [list2vec(v) for v in [[1,1,0],[0,1,1],[1,0,1]]]
>>> morph(S, B)
[(Vec({0, 1, 2},{0: 1, 1: 1, 2: 0}), Vec({0, 1, 2},{0: 1, 1: 0, 2: 0})), (Vec({0, 1, 2},{0: 0, 1: 1, 2: 1}), Vec({0, 1, 2},{0: 0, 1: 1, 2: 0})), (Vec({0, 1, 2},{0: 1, 1: 0, 2: 1}), Vec({0, 1, 2},{0: 0, 1: 0, 2: 1}))]
'''
A = []
Sprime = S[:]
ret = []
while len(A) < len(B):
for vec in B:
A.append(vec)
if is_independent(A):
A.pop()
ejected = exchange(Sprime, A, vec)
Sprime[Sprime.index(ejected)] = vec
ret.append((vec, ejected))
A.append(vec)
else:
A.pop()
return ret
def morph(S, B):
'''
Input:
- S: a list of distinct Vec instances
- B: a list of linearly independent Vec instances
- Span S == Span B
Output: a list of pairs of vectors to inject and eject
Example:
>>> #This is how our morph works. Yours may yield different results.
>>> S = [list2vec(v) for v in [[1,0,0],[0,1,0],[0,0,1]]]
>>> B = [list2vec(v) for v in [[1,1,0],[0,1,1],[1,0,1]]]
>>> morph(S, B)
[(Vec({0, 1, 2},{0: 1, 1: 1, 2: 0}), Vec({0, 1, 2},{0: 1, 1: 0, 2: 0})), (Vec({0, 1, 2},{0: 0, 1: 1, 2: 1}), Vec({0, 1, 2},{0: 0, 1: 1, 2: 0})), (Vec({0, 1, 2},{0: 1, 1: 0, 2: 1}), Vec({0, 1, 2},{0: 0, 1: 0, 2: 1}))]
'''
A = []
Sprime = S[:]
ret = []
while len(A) < len(B):
for vec in B:
A.append(vec)
if is_independent(A):
A.pop()
ejected = exchange(Sprime, A, vec)
Sprime[Sprime.index(ejected)] = vec
ret.append((vec, ejected))
A.append(vec)
else:
A.pop()
return ret
def is_invertible(M):
'''
input: A matrix, M
outpit: A boolean indicating if M is invertible.
>>> M = Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): 0, (1, 2): 1, (3, 2): 0, (0, 0): 1, (3, 3): 4, (3, 0): 0, (3, 1): 0, (1, 1): 2, (2, 1): 0, (0, 2): 1, (2, 0): 0, (1, 3): 0, (2, 3): 1, (2, 2): 3, (1, 0): 0, (0, 3): 0})
>>> is_invertible(M)
True
'''
return len(M.D[0]) == len(M.D[1]) and is_independent([v for k,v in mat2coldict(M).items() ])
def subset_basis(T):
'''
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
'''
while not is_independent(T):
T.pop()
return T
def subset_basis(T):
'''
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
'''
while not is_independent(T):
T.pop()
return T
def subset_basis(T):
"""
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
"""
S = list(T)
for i in range(len(T)):
if is_independent(S):
return S
S.remove(S[i])
def is_invertible(M):
'''
input: A matrix, M
outpit: A boolean indicating if M is invertible.
>>> M = Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): 0, (1, 2): 1, (3, 2): 0, (0, 0): 1, (3, 3): 4, (3, 0): 0, (3, 1): 0, (1, 1): 2, (2, 1): 0, (0, 2): 1, (2, 0): 0, (1, 3): 0, (2, 3): 1, (2, 2): 3, (1, 0): 0, (0, 3): 0})
>>> is_invertible(M)
True
'''
from hw4 import is_independent
from matutil import mat2rowdict, mat2coldict
rowsdict = mat2rowdict(M)
rowslist = [ rowsdict[i] for i in rowsdict.keys() ]
coldict = mat2coldict(M)
collist = [ coldict[i] for i in coldict.keys() ]
if len(rowslist) == len(collist) and is_independent(collist): return True
else: return False
def subset_basis(T):
'''
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
'''
from hw4 import is_independent,is_superfluous
T_temp = list(T)
if not is_independent(T):
for i in range(len(T)-1,-1,-1):
if is_superfluous(T_temp,i): T_temp.remove(T[i])
return T_temp
def subset_basis(T):
'''
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
'''
from hw4 import is_independent
basis = list()
for vec in T:
if True == is_independent(basis+[vec]):
basis.append(vec)
return basis
def subset_basis(T):
'''
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
'''
C=[]
for i,v in enumerate(T):
C.append(v)
if not is_independent(C):
C.pop()
return C
def is_invertible(M):
'''
input: A matrix, M
outpit: A boolean indicating if M is invertible.
>>> M = Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): 0, (1, 2): 1, (3, 2): 0, (0, 0): 1, (3, 3): 4, (3, 0): 0, (3, 1): 0, (1, 1): 2, (2, 1): 0, (0, 2): 1, (2, 0): 0, (1, 3): 0, (2, 3): 1, (2, 2): 3, (1, 0): 0, (0, 3): 0})
>>> is_invertible(M)
True
'''
from hw4 import is_independent
from matutil import mat2rowdict, mat2coldict
rowsdict = mat2rowdict(M)
rowslist = [rowsdict[i] for i in rowsdict.keys()]
coldict = mat2coldict(M)
collist = [coldict[i] for i in coldict.keys()]
if len(rowslist) == len(collist) and is_independent(collist): return True
else: return False
def subset_basis(T):
'''
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
'''
from hw4 import is_independent, is_superfluous
T_temp = list(T)
if not is_independent(T):
for i in range(len(T) - 1, -1, -1):
if is_superfluous(T_temp, i): T_temp.remove(T[i])
return T_temp
def subset_basis(T):
'''
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
'''
rank_T = rank(T)
S = []
for x in T:
if(is_independent([S+[x]])):
S.append(x)
if rank(S) == rank_T:break
return S
def subset_basis(T):
"""
input: A list, T, of Vecs
output: A list, S, containing Vecs from T, that is a basis for the
space spanned by T.
>>> 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})
>>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
True
"""
from hw4 import is_independent
basis = list()
for vec in T:
if True == is_independent(basis + [vec]):
basis.append(vec)
return basis