def solve(A, b):
M = echelon.transformation(A)
U = M * A
col_label_list = sorted(A.D[1])
U_rows_dict = mat2rowdict(U)
row_list = [U_rows_dict[i] for i in sorted(U_rows_dict)]
return echelon_solve(row_list, col_label_list, M * b)
def solve2(A, b):
M = transformation(A)
U = M * A
U_rowdict = mat2rowdict(U)
rowlist = [U_rowdict[i] for i in sorted(U_rowdict)]
label_list = (A.D[1])
return echelon_solve(rowlist, label_list, M * b)
def solve(A, b):
M = echelon.transformation(A)
U = M*A
col_label_list = sorted(A.D[1])
U_rows_dict = mat2rowdict(U)
rowlist = [U_rows_dict[i] for i in U_rows_dict]
return echelon_solve(rowlist,col_label_list, M*b)
def solve(A, b):
col_label_list = sorted(A.D[1])
M = echelon.transformation(A, col_label_list)
U = M*A
U_rows_dict = mat2rowdict(U)
rowlist = [U_rows_dict[i] for i in U_rows_dict]
x = M*b
return echelon_solve(rowlist, col_label_list, [x[item] for item in x.D])
def basis_AT(A):
basis = []
M = transformation(A)
U = M * A
M_rowdict = mat2rowdict(M)
U_rowdict = mat2rowdict(U)
M_rowlist = [M_rowdict[i] for i in sorted(M_rowdict)]
U_rowlist = [U_rowdict[i] for i in sorted(U_rowdict)]
zero_vector = zero_vec(U_rowlist[0].D)
for i in range(len(U_rowlist)):
if (U_rowlist[i] == zero_vector):
basis.append(M_rowlist[i])
return basis
def echelon_solve(rowlist, label_list, b):
'''
Input:
- rowlist: a list of Vecs
- label_list: a list of labels establishing an order on the domain of
Vecs in rowlist
- b: a vector (represented as a list)
Output:
- Vec x such that rowlist * x is b
>>> D = {'A','B','C','D','E'}
>>> U_rows = [Vec(D, {'A':one, 'E':one}), Vec(D, {'B':one, 'E':one}), Vec(D,{'C':one})]
>>> b_list = [one,0,one]>>> cols = ['A', 'B', 'C', 'D', 'E']
>>> echelon_solve(U_rows, cols, b_list)
Vec({'B', 'C', 'A', 'D', 'E'},{'B': 0, 'C': one, 'A': one})
'''
A=rowdict2mat(rowlist)
M=echelon.transformation(A)
U=M*A
rowlist=mat2rowdict(U)
temp=M*list2vec(b)
res = []
for label in range(len(b)):
if label in temp.f and temp.f[label]:
res.append(one)
else:
res.append(0)
b=res
result = Vec(set(label_list), {})
counter = len(b)-1
for row in reversed(list(rowlist.values())):
for label in label_list:
if label in row.f and row.f[label]!=0:
result.f[label] = b[counter]-row*result
break
counter = counter -1
return result
print(gcd(23, 15))
N = 367160330145890434494322103
a = 67469780066325164
b = 9429601150488992
c = a * a - b * b
print(gcd(a - b, N))
print(make_Vec({2, 3, 5, 7, 11}, [(3, 1)]))
print(make_Vec({2, 3, 5, 7, 11}, [(2, 17), (3, 0), (5, 1), (11, 3)]))
print(find_candidates(2419, primes(32))[1])
A = listlist2mat([[0, 2, 3, 4, 5], [0, 0, 0, 3, 2], [1, 2, 3, 4, 5],
[0, 0, 0, 6, 7], [0, 0, 0, 9, 8]])
print(A)
M = transformation(A)
print(M)
# print(M * A)
print(M * A)
# print()
# print(listlist2mat([[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 0, 2, 1, 0], [0, 0, 3.004, 0.667, 1]]) * (M * A))
print(
row_reduce([
list2vec([one, one, 0, 0]),
list2vec([one, 0, one, 0]),
list2vec([0, one, one, one]),
list2vec([one, 0, 0, 0])
]))
# *********************** PROBLEMS *************************
U_rows_dict = mat2rowdict(U)
rowlist = [U_rows_dict[i] for i in U_rows_dict]
x = M*b
return echelon_solve(rowlist, col_label_list, [x[item] for item in x.D])
D = {'A', 'B', 'C', 'D'}
A = Mat(({'a', 'b', 'c', 'd'}, {'A', 'B', 'C', 'D'}),
{('a', 'B'): one, ('a', 'C'): 0, ('d', 'C'): one, ('a', 'A'): one, ('d', 'D'): one, ('d', 'A'): 0,
('d', 'B'): 0, ('b', 'B'): 0, ('c', 'B'): one, ('a', 'C'): 0, ('c', 'A'): one, ('b', 'D'): one,
('c', 'D'): one, ('c', 'C'): one, ('b', 'A'): one, ('a', 'D'): one})
r = Vec({'a', 'b', 'c', 'd'}, {'a': one, 'c': one})
col_label_list = sorted(A.D[1])
M = echelon.transformation(A)
U = M*A
U_rows_dict = mat2rowdict(U)
x = M*r
rowlist = [U_rows_dict[i] for i in U_rows_dict]
label_list = col_label_list
b = [x[item] for item in x.D]
## Problem 7
null_space_rows_a = {3, 4} # Put the row numbers of M from the PDF
