def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
return [vlist[i] for i in range(len(vlist)) if orthogonalize(vlist)[i] * orthogonalize(vlist)[i] > 1e-20]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
return [ (1/math.sqrt(x*x)) * x for x in orthogonalize(L)]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
list1 = [[4, 3, 1, 2], [8, 9, -5, -5], [10, 1, -1, 5]]
for i in list1:
list2vec(i)
>>> L = [Vec({0, 1, 2, 3},{0: 4, 1: 3, 2: 1, 3: 2}),Vec({0, 1, 2, 3},{0: 8, 1: 9, 2: -5, 3: -5}), Vec({0, 1, 2, 3},{0: 10, 1: 1, 2: -1, 3: 5}) ]
>>> from vecutil import *
>>> v1 = list2vec([1,2,3,4])
>>> v2 = list2vec([2,3,4,5])
>>> print(test_format(orthonormalization.orthonormalize([v1, v2])))
[Vec({0.000000, 1.000000, 2.000000, 3.000000}, {0.000000: 0.182574, 1.000000: 0.365148, 2.000000: 0.547723, 3.000000: 0.730297}), Vec({0.000000, 1.000000, 2.000000, 3.000000}, {0.000000: 0.816497, 1.000000: 0.408248, 3.000000: -0.408248})]
>>> v1 = list2vec([1,2,1])
>>> v2 = list2vec([2,2,2])
>>> print(test_format(orthonormalization.orthonormalize([v1, v2])))
[Vec({0.000000, 1.000000, 2.000000}, {0.000000: 0.408248, 1.000000: 0.816497, 2.000000: 0.408248}), Vec({0.000000, 1.000000, 2.000000}, {0.000000: 0.577350, 1.000000: -0.577350, 2.000000: 0.577350})]
>>> v1 = list2vec([.2, .4, .1, .9])
>>> v2 = list2vec([12, 144, 91, 0])
>>> v3 = list2vec([1, 1, 1, 1])
>>> v4 = list2vec([0, 12, 0, 0])
>>> print(test_format(orthonormalization.orthonormalize([v1, v2, v3, v4])))
[Vec({0.000000, 1.000000, 2.000000, 3.000000}, {0.000000: 0.198030, 1.000000: 0.396059, 2.000000: 0.099015, 3.000000: 0.891133}), Vec({0.000000, 1.000000, 2.000000, 3.000000}, {0.000000: -0.009900, 1.000000: 0.747167, 2.000000: 0.538319, 3.000000: -0.389687}), Vec({0.000000, 1.000000, 2.000000, 3.000000}, {0.000000: 0.827563, 1.000000: -0.344536, 2.000000: 0.436070, 3.000000: -0.079228}), Vec({0.000000, 1.000000, 2.000000, 3.000000}, {0.000000: 0.525190, 1.000000: 0.407644, 2.000000: -0.714319, 3.000000: -0.218515})]
'''
Ortho_L = orthogonalize(L)
list1 = []
for i in Ortho_L:
list1.append(sqrt(i*i))
return [v/vnorm for v, vnorm in zip(Ortho_L, list1)]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list Lstar of len(L) orthonormal Vecs such that, for all i in range(len(L)),
Span L[:i+1] == Span Lstar[:i+1]
>>> from vec import Vec
>>> D = {'a','b','c','d'}
>>> L = [Vec(D, {'a':4,'b':3,'c':1,'d':2}), Vec(D, {'a':8,'b':9,'c':-5,'d':-5}), Vec(D, {'a':10,'b':1,'c':-1,'d':5})]
>>> for v in orthonormalize(L): print(v)
...
<BLANKLINE>
a b c d
-----------------------
0.73 0.548 0.183 0.365
<BLANKLINE>
a b c d
--------------------------
0.187 0.403 -0.566 -0.695
<BLANKLINE>
a b c d
--------------------------
0.528 -0.653 -0.512 0.181
'''
vstarlist = orthogonalization.orthogonalize(
L) #obrain V* veclist and matrix T
return [normalize(v)
for v in vstarlist] # obtain the Q matrix as a list of vecs
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list Lstar of len(L) orthonormal Vecs such that, for all i in range(len(L)),
Span L[:i+1] == Span Lstar[:i+1]
>>> from vec import Vec
>>> D = {'a','b','c','d'}
>>> L = [Vec(D, {'a':4,'b':3,'c':1,'d':2}), Vec(D, {'a':8,'b':9,'c':-5,'d':-5}), Vec(D, {'a':10,'b':1,'c':-1,'d':5})]
>>> for v in orthonormalize(L): print(v)
...
<BLANKLINE>
a b c d
-----------------------
0.73 0.548 0.183 0.365
<BLANKLINE>
a b c d
--------------------------
0.187 0.403 -0.566 -0.695
<BLANKLINE>
a b c d
--------------------------
0.528 -0.653 -0.512 0.181
'''
return [v/sqrt(v*v) for v in orthogonalize(L) if 1e-50 < v*v]
def orthonormalize(L):
"""
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
"""
return list(map(lambda v: (1.0 / sqrt(v * v)) * v if v * v > 1e-20 else v, orthogonalize(L)))
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list Lstar of len(L) orthonormal Vecs such that, for all i in range(len(L)),
Span L[:i+1] == Span Lstar[:i+1]
>>> from vec import Vec
>>> D = {'a','b','c','d'}
>>> L = [Vec(D, {'a':4,'b':3,'c':1,'d':2}), Vec(D, {'a':8,'b':9,'c':-5,'d':-5}), Vec(D, {'a':10,'b':1,'c':-1,'d':5})]
>>> for v in orthonormalize(L): print(v)
...
<BLANKLINE>
a b c d
-----------------------
0.73 0.548 0.183 0.365
<BLANKLINE>
a b c d
--------------------------
0.187 0.403 -0.566 -0.695
<BLANKLINE>
a b c d
--------------------------
0.528 -0.653 -0.512 0.181
'''
Lstar = orthogonalize(L)
return [(1/sqrt(v*v)) * v for v in Lstar if not v.is_almost_zero()]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
return [normalize(v) for v in orthogonalize(L)]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
>>> from vec import Vec
>>> D = {'a','b','c','d'}
>>> L = [Vec(D, {'a':4,'b':3,'c':1,'d':2}), Vec(D, {'a':8,'b':9,'c':-5,'d':-5}), Vec(D, {'a':10,'b':1,'c':-1,'d':5})]
>>> for v in orthonormalize(L): print(v)
...
<BLANKLINE>
b a c d
-----------------------
0.548 0.73 0.183 0.365
<BLANKLINE>
b a c d
--------------------------
0.403 0.187 -0.566 -0.695
<BLANKLINE>
b a c d
--------------------------
-0.653 0.528 -0.512 0.181
'''
return ( i / sqrt(i**2) for i in orthogonalize(L))
def orthonormalize(L):
"""
Input: a list L of linearly independent Vecs
Output: A list Lstar of len(L) orthonormal Vecs such that, for all i in range(len(L)),
Span L[:i+1] == Span Lstar[:i+1]
>>> from vec import Vec
>>> D = {'a','b','c','d'}
>>> L = [Vec(D, {'a':4,'b':3,'c':1,'d':2}), Vec(D, {'a':8,'b':9,'c':-5,'d':-5}), Vec(D, {'a':10,'b':1,'c':-1,'d':5})]
>>> for v in orthonormalize(L): print(v)
...
<BLANKLINE>
a b c d
-----------------------
0.73 0.548 0.183 0.365
<BLANKLINE>
a b c d
--------------------------
0.187 0.403 -0.566 -0.695
<BLANKLINE>
a b c d
--------------------------
0.528 -0.653 -0.512 0.181
"""
vstarlist = orthogonalization.orthogonalize(L) # obrain V* veclist and matrix T
return [normalize(v) for v in vstarlist] # obtain the Q matrix as a list of vecs
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
return [v / sqrt(v * v) for v in orthogonalize(L)]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [u for u in orthogonalize(vlist) if u * u > 10E-20]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [i for i in orthogonalize(vlist) if i * i > (10**-20)**0.5]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [iVec for iVec in orthogonalize(vlist) if iVec * iVec > 10**-20]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return list(filter(lambda v : v*v > 1e-20, orthogonalize(vlist)))
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [ i for i in orthogonalize(vlist) if not(abs(i * i) < 10**(-20)) ]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [v for v in orthogonalize(vlist) if v*v >1e-11]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
return [ vlist[i] for i,x in enumerate(orthogonalize(vlist)) if x*x>1e-20 ]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [ x for x in orthogonalize(vlist) if x*x>1e-20 ]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
return [vlist[k] for k, v in enumerate(orthogonalize(vlist)) if square_norm(v) > 1E-20]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return list(filter(lambda v: v * v > 1e-20, orthogonalize(vlist)))
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [v for v in orthogonalize(vlist) if v * v > 1e-11]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [x for x in orthogonalize(vlist) if x * x > 10**-20]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [v for v in orthogonalize(vlist) if square_norm(v) > 1E-20]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
olist = orthogonalize(L)
return [ o/sqrt(o*o) for o in olist]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [e for e in orthogonalize(vlist) if e * e > 1e-20]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [v for v in orthogonalize(vlist) if square_norm(v) > 1E-20]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [ i for i in orthogonalize(vlist) if i*i>(10**-20)**0.5 ]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
from vec import dot
return [v for v in orthogonalize(vlist) if dot(v,v) > 1e-20]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
v_star = orthogonalize(vlist)
return [vlist[v] for v in range(len(vlist)) if v_star[v]*v_star[v] > 10e-20]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
O = orthogonalize(vlist)
return [vlist[x] for x in range(len(vlist)) if O[x] * O[x] > 10**-20]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
L_star = orthogonalize(L)
L_norm = [ sqrt(v*v) for v in L_star ]
return [ L_star[i]/L_norm[i] for i in range(len(L_star)) ]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
from orthogonalization import orthogonalize
oVecs = orthogonalize(L)
return [v/(v*v)**(1/2) for v in oVecs]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
O = orthogonalize(vlist)
return [vlist[x] for x in range(len(vlist)) if O[x]*O[x] >10**-20 ]
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
olist = orthogonalize(vlist)
return [o for o in olist if o*o>1e-20]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
orth_list = orthogonalize(vlist)
return [v for i, v in enumerate(vlist) if orth_list[i] * orth_list[i] > 10 ** -20]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
o = orthogonalize(L)
l = [sqrt(square_norm(v)) for v in o]
return [v/l[k] for k, v in enumerate(o)]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
orthoL = orthogonalize(L)
normL = [sqrt(li * li) for li in orthoL]
return [li / ni for li, ni in zip(orthoL, normL)]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
orthoL = orthogonalize(L)
normL = [sqrt(li*li) for li in orthoL]
return [li/ni for li,ni in zip(orthoL, normL)]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
v_star = orthogonalize(vlist)
return [ vlist[i] for i in range(len(vlist)) if not( abs(v_star[i] * v_star[i]) < 10**(-20) ) ]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
l = orthogonalize(L)
norms = [sqrt(x*x) for x in l]
return [l/n for n,l in zip(norms, l)]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist[v]
'''
from vec import dot
return [y for (x,y) in zip(orthogonalize(vlist),vlist) if dot(x,x)>1e-20]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
orths = orthogonalize(L)
norms = [sqrt(v*v) for v in orths]
return [(1/norms[i])*vec for i, vec in enumerate(orths)]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
return list(
map(lambda v: (1.0 / sqrt(v * v)) * v if v * v > 1e-20 else v,
orthogonalize(L)))
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
o = orthogonalize(L)
l = [sqrt(square_norm(v)) for v in o]
return [v / l[k] for k, v in enumerate(o)]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
return [
vlist[i] for i, e in enumerate(orthogonalize(vlist)) if e * e > 1e-20
]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
aa = []
o = orthogonalize(vlist)
return [vlist[i] for i, o in enumerate(o) if o * o > 1e-20]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
return [
vlist[k] for k, v in enumerate(orthogonalize(vlist))
if square_norm(v) > 1E-20
]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
orth_list = orthogonalize(vlist)
return [
v for i, v in enumerate(vlist) if orth_list[i] * orth_list[i] > 10**-20
]
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
from orthogonalization import orthogonalize
from math import sqrt
L_star = orthogonalize(L)
return [v / sqrt(v * v) for v in L_star]
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
oVecs = orthogonalize(vlist)
vecs = vlist[:]
for i in range(len(oVecs)):
norm = sum([oVecs[i][k]**2 for k in oVecs[i].D])**(1 / 2)
if norm**2 <= 10**-20:
vecs.remove(vlist[i])
return vecs
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
r = []
i = 0
for v in orthogonalize(vlist):
if (v*v) > 1e-12:
r.append(vlist[i])
i = i+1
return r
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
orthoVecs = orthogonalize(vlist)
basis = []
for v in orthoVecs:
norm = sum([v[k]**2 for k in v.D])**(1 / 2)
if norm**2 > 10**-20:
basis.append(v)
return basis
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list T of orthonormal Vecs such that for all i in [1, len(L)],
Span L[:i] == Span T[:i]
'''
L1 = orthogonalize(L)
L2 = []
for i in L1:
temp = 0
for j in i.D:
temp += i[j] ** 2
L2.append(temp)
return [L1[i] / sqrt(L2[i]) for i in range(len(L1))]
def aug_orthonormalize(L):
'''
Input:
- L: a list of Vecs
Output:
- A pair Qlist, Rlist such that:
* coldict2mat(L) == coldict2mat(Qlist) * coldict2mat(Rlist)
* Qlist = orthonormalize(L)
'''
def adjust(v,multipliers):
return Vec(v.D,{i: v.f[i]*multipliers[i] if i in v.f else 0 for i in v.D})
q,r = aug_orthogonalize(L)
Qlist = orthonormalize(q)
Rlist = [ adjust(v,[ sqrt(v*v) for v in orthogonalize(L) ]) for v in r ]
return Qlist,Rlist
def subset_basis(vlist):
return [
vlist[i] for i, e in enumerate(orthogonalize(vlist)) if e * e > 1e-20
]
def basis(vlist):
a = orthogonalize(vlist)
#print(np.linalg.norm(list(a[0].f.values()),2))
return [e for e in orthogonalize(vlist) if e * e > 1e-20]
def main():
# Constants for Now - Should Read In
t = 1
U = 0.0
size = 4
delta_tau = .01
n_up = 2
n_down = 2
total_steps = 10000
steps_orthog = 10
steps_measure = 20
# Matrices Needed Throughout Simulation
one_body_matrix = np.zeros((size, size))
one_body_propagator = np.zeros((size, size))
neighbors = np.zeros((size, 2))
trial_wf_up = np.zeros((size, n_up))
trial_wf_down = np.zeros((size, n_down))
wf_up = np.zeros((size, n_up))
wf_down = np.zeros((size, n_down))
# Form the One Body Portion of the Hamiltonian
ob.form_one_body_matrix(one_body_matrix, neighbors, size, t)
# Exponentiate That One Body Portion of the Hamiltonian So You Can Propagate
ob.exponentiate_one_body(one_body_propagator, one_body_matrix, trial_wf_up,
trial_wf_down, size, n_up, n_down, delta_tau)
# Copy Trial Wave Function to Current Wave Function For Propagattion Loop
wf_up = trial_wf_up
wf_down = trial_wf_down
# Propagate By Looping Over Products of One and Two Body Propagators
for steps in range(total_steps):
# First Propagate by Half of One Body Term
ob.propagate_one_body(one_body_propagator, wf_up, wf_down, size, n_up,
n_down)
# Then Propagate By Two Body Term
tb.propagate_two_body(wf_up, wf_down, size, n_up, n_down, U, delta_tau)
# Lastly Propagate by Half of One Body Term
ob.propagate_one_body(one_body_propagator, wf_up, wf_down, size, n_up,
n_down)
# Orthogonalize the Wave Functions to Ensure No Collapse
if (steps % steps_orthog == 0):
og.orthogonalize(wf_up, wf_down, size, n_up, n_down)
# Measure Observables Like the Energy Every So Often
if (steps % steps_measure == 0):
current_energy = ms.measure_total_energy(wf_up, wf_down,
trial_wf_up,
trial_wf_down, neighbors,
size, n_up, n_down, U, t)
print current_energy
# Append File with Values
f = open("energy.dat", "a+")
f.write("%i %5.2f\n" % (steps, current_energy))
f.close()
ortho_compl_generators_2 = [project_orthogonal(v, U_vecs_2) for v in W_vecs_2]
U_vecs_3 = [list2vec(v) for v in [[-4,3,1,-2],[-2,2,3,-1]]]
W_vecs_3 = [list2vec(v) for v in [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]]
# Give a list of Vecs
ortho_compl_generators_3 = [project_orthogonal(v, U_vecs_3) for v in W_vecs_3]
## 2: (Problem 9.11.3) Basis for null space
# Your solution should be a list of Vecs
rows_A = [list2vec(v) for v in [[-4,-1,-3,-2], [0,4,0,-1]]]
r_basis = [list2vec(v) for v in [[1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]]]
L = rows_A + r_basis
Lstar = orthogonalize(L)
null_space_basis = [v for v in Lstar if not v.is_almost_zero()]
## 3: (Problem 9.11.9) Orthonormalize(L)
def orthonormalize(L):
'''
Input: a list L of linearly independent Vecs
Output: A list Lstar of len(L) orthonormal Vecs such that, for all i in range(len(L)),
Span L[:i+1] == Span Lstar[:i+1]
>>> from vec import Vec
>>> D = {'a','b','c','d'}
>>> L = [Vec(D, {'a':4,'b':3,'c':1,'d':2}), Vec(D, {'a':8,'b':9,'c':-5,'d':-5}), Vec(D, {'a':10,'b':1,'c':-1,'d':5})]
