[razvan@ohio data]$ python3
Python 3.8.5 (default, Sep  4 2020, 02:22:02) 
[Clang 10.0.0 ] :: Anaconda, Inc. on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> import numpy as np
>>> a = np.array([1, 2, 3, 4])
>>> a
array([1, 2, 3, 4])
>>> a.shape
(4,)
>>> a.ndim
1
>>> len(a.shape)
1
>>> list(range(8))
[0, 1, 2, 3, 4, 5, 6, 7]
>>> a = np.array(list(range(8)))
>>> a
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> b = np.arange(8)
>>> b
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> a = np.array([1, 2, 3, 4])
>>> a
array([1, 2, 3, 4])
>>> b
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> b.shape
(8,)
>>> b.reshape(4,2)
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> b
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> b = b.reshape(4,2)
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> a
array([1, 2, 3, 4])
>>> b.shape
(4, 2)
>>> b.ndim
2
>>> np.column_stack((a, b))
array([[1, 0, 1],
       [2, 2, 3],
       [3, 4, 5],
       [4, 6, 7]])
>>> test = np.loadtxt('test.txt')
>>> test.shape
(31, 2)
>>> test[:5]
array([[  2227., 356700.],
       [  2596., 233000.],
       [  2476., 233000.],
       [  2556., 299000.],
       [  2391., 325000.]])
>>> test[:5,1]
array([356700., 233000., 233000., 299000., 325000.])
>>> test[:5,0]
array([2227., 2596., 2476., 2556., 2391.])
>>> test[:5,:]
array([[  2227., 356700.],
       [  2596., 233000.],
       [  2476., 233000.],
       [  2556., 299000.],
       [  2391., 325000.]])
>>> t = test[:,1]
>>> t[:5]
array([356700., 233000., 233000., 299000., 325000.])
>>> t.shape
(31,)
>>> X = test[:,0]
>>> X[:5]
array([2227., 2596., 2476., 2556., 2391.])
>>> w = np.array([1, 100])
>>> w[0]
1
>>> w[1]
100
>>> X.size
31
>>> X.shape[1]
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
IndexError: tuple index out of range
>>> X.shape[0]
31
>>> ones = np.ones(X.shape[0])
>>> ones[:5]
array([1., 1., 1., 1., 1.])
>>> np.column_stack((ones,X))
array([[1.000e+00, 2.227e+03],
       [1.000e+00, 2.596e+03],
       [1.000e+00, 2.476e+03],
       [1.000e+00, 2.556e+03],
       [1.000e+00, 2.391e+03],
       [1.000e+00, 1.580e+03],
       [1.000e+00, 3.157e+03],
       [1.000e+00, 3.088e+03],
       [1.000e+00, 1.568e+03],
       [1.000e+00, 2.774e+03],
       [1.000e+00, 1.888e+03],
       [1.000e+00, 1.800e+03],
       [1.000e+00, 1.876e+03],
       [1.000e+00, 1.800e+03],
       [1.000e+00, 1.614e+03],
       [1.000e+00, 1.568e+03],
       [1.000e+00, 2.425e+03],
       [1.000e+00, 2.462e+03],
       [1.000e+00, 3.282e+03],
       [1.000e+00, 2.741e+03],
       [1.000e+00, 2.656e+03],
       [1.000e+00, 2.606e+03],
       [1.000e+00, 1.506e+03],
       [1.000e+00, 1.614e+03],
       [1.000e+00, 4.224e+03],
       [1.000e+00, 3.000e+03],
       [1.000e+00, 1.884e+03],
       [1.000e+00, 3.170e+03],
       [1.000e+00, 3.396e+03],
       [1.000e+00, 2.738e+03],
       [1.000e+00, 3.712e+03]])
>>> newX = np.column_stack((ones,X))
>>> newX.shape
(31, 2)
>>> w
array([  1, 100])
>>> newX.dot(w)
array([222701., 259601., 247601., 255601., 239101., 158001., 315701.,
       308801., 156801., 277401., 188801., 180001., 187601., 180001.,
       161401., 156801., 242501., 246201., 328201., 274101., 265601.,
       260601., 150601., 161401., 422401., 300001., 188401., 317001.,
       339601., 273801., 371201.])
>>> @
  File "<stdin>", line 1
    @
    ^
SyntaxError: invalid syntax
>>> newX @ w
array([222701., 259601., 247601., 255601., 239101., 158001., 315701.,
       308801., 156801., 277401., 188801., 180001., 187601., 180001.,
       161401., 156801., 242501., 246201., 328201., 274101., 265601.,
       260601., 150601., 161401., 422401., 300001., 188401., 317001.,
       339601., 273801., 371201.])
>>> newX * w
array([[1.000e+00, 2.227e+05],
       [1.000e+00, 2.596e+05],
       [1.000e+00, 2.476e+05],
       [1.000e+00, 2.556e+05],
       [1.000e+00, 2.391e+05],
       [1.000e+00, 1.580e+05],
       [1.000e+00, 3.157e+05],
       [1.000e+00, 3.088e+05],
       [1.000e+00, 1.568e+05],
       [1.000e+00, 2.774e+05],
       [1.000e+00, 1.888e+05],
       [1.000e+00, 1.800e+05],
       [1.000e+00, 1.876e+05],
       [1.000e+00, 1.800e+05],
       [1.000e+00, 1.614e+05],
       [1.000e+00, 1.568e+05],
       [1.000e+00, 2.425e+05],
       [1.000e+00, 2.462e+05],
       [1.000e+00, 3.282e+05],
       [1.000e+00, 2.741e+05],
       [1.000e+00, 2.656e+05],
       [1.000e+00, 2.606e+05],
       [1.000e+00, 1.506e+05],
       [1.000e+00, 1.614e+05],
       [1.000e+00, 4.224e+05],
       [1.000e+00, 3.000e+05],
       [1.000e+00, 1.884e+05],
       [1.000e+00, 3.170e+05],
       [1.000e+00, 3.396e+05],
       [1.000e+00, 2.738e+05],
       [1.000e+00, 3.712e+05]])
>>> a
array([1, 2, 3, 4])
>>> a * a
array([ 1,  4,  9, 16])
>>> c = np.array([1, -1, 2, -2])
>>> a * c
array([ 1, -2,  6, -8])
>>> a
array([1, 2, 3, 4])
>>> c
array([ 1, -1,  2, -2])
>>> 2 * a
array([2, 4, 6, 8])
>>> d = np.array([2, 2, 2, 2])
>>> d * a
array([2, 4, 6, 8])
>>> a / 2
array([0.5, 1. , 1.5, 2. ])
>>> a / 2
array([0.5, 1. , 1.5, 2. ])
>>> w
array([  1, 100])
>>> w * w
array([    1, 10000])
>>> np.sum(w * w)
10001
>>> w.dot(w)
10001
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> d
array([2, 2, 2, 2])
>>> d[0] = 1
>>> d[2] = -1
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> d
array([ 1,  2, -1,  2])
>>> b.shape
(4, 2)
>>> d.shape
(4,)
>>> d.dot(b)
array([12, 16])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> b.ravel()
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> b.ravel(order = 'F')
array([0, 2, 4, 6, 1, 3, 5, 7])
>>> b.ravel(order = 'C')
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> v = b.ravel(order = 'C')
>>> v
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> v.reshape(4, -1)
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> v.reshape(-1, -1)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: can only specify one unknown dimension
>>> v.reshape(-1, 2)
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> v
array([0, 1, 2, 3, 4, 5, 6, 7])
>>> d
array([ 1,  2, -1,  2])
>>> from np import newaxis
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ModuleNotFoundError: No module named 'np'
>>> from numpy import newaxis
>>> d[:,newaxis]
array([[ 1],
       [ 2],
       [-1],
       [ 2]])
>>> d[:,newaxis].shape
(4, 1)
>>> d[newaxis,:]
array([[ 1,  2, -1,  2]])
>>> d[newaxis,:].shape
(1, 4)
>>> d.reshape(4,-1)
array([[ 1],
       [ 2],
       [-1],
       [ 2]])
>>> np.ones(10)
array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1.])
>>> np.zeros(10)
array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
>>> np.eye(10)
array([[1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
       [0., 1., 0., 0., 0., 0., 0., 0., 0., 0.],
       [0., 0., 1., 0., 0., 0., 0., 0., 0., 0.],
       [0., 0., 0., 1., 0., 0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 1., 0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0., 1., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0., 0., 1., 0., 0., 0.],
       [0., 0., 0., 0., 0., 0., 0., 1., 0., 0.],
       [0., 0., 0., 0., 0., 0., 0., 0., 1., 0.],
       [0., 0., 0., 0., 0., 0., 0., 0., 0., 1.]])
>>> np.eye(10) + 10
array([[11., 10., 10., 10., 10., 10., 10., 10., 10., 10.],
       [10., 11., 10., 10., 10., 10., 10., 10., 10., 10.],
       [10., 10., 11., 10., 10., 10., 10., 10., 10., 10.],
       [10., 10., 10., 11., 10., 10., 10., 10., 10., 10.],
       [10., 10., 10., 10., 11., 10., 10., 10., 10., 10.],
       [10., 10., 10., 10., 10., 11., 10., 10., 10., 10.],
       [10., 10., 10., 10., 10., 10., 11., 10., 10., 10.],
       [10., 10., 10., 10., 10., 10., 10., 11., 10., 10.],
       [10., 10., 10., 10., 10., 10., 10., 10., 11., 10.],
       [10., 10., 10., 10., 10., 10., 10., 10., 10., 11.]])
>>> np.ones(5)
array([1., 1., 1., 1., 1.])
>>> np.ones(4, 2)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/Users/razvan/opt/anaconda3/lib/python3.8/site-packages/numpy/core/numeric.py", line 192, in ones
    a = empty(shape, dtype, order)
TypeError: Cannot interpret '2' as a data type
>>> np.ones((4, 2))
array([[1., 1.],
       [1., 1.],
       [1., 1.],
       [1., 1.]])
>>> np.zeros((4, 2))
array([[0., 0.],
       [0., 0.],
       [0., 0.],
       [0., 0.]])
>>> np.ones((4, 2)) * 2
array([[2., 2.],
       [2., 2.],
       [2., 2.],
       [2., 2.]])
>>> d
array([ 1,  2, -1,  2])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> v = np.ones(4)
>>> v
array([1., 1., 1., 1.])
>>> vv = no.ones((2,4))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
NameError: name 'no' is not defined
>>> vv = np.ones((2,4))
>>> vv
array([[1., 1., 1., 1.],
       [1., 1., 1., 1.]])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> b + vv
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (4,2) (2,4) 
>>> b + vv.T
array([[1., 2.],
       [3., 4.],
       [5., 6.],
       [7., 8.]])
>>> vv.T.shape
(4, 2)
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> v
array([1., 1., 1., 1.])
>>> b + v
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (4,2) (4,) 
>>> v
array([1., 1., 1., 1.])
>>> v.shape
(4,)
>>> b + v[:,newaxis]
array([[1., 2.],
       [3., 4.],
       [5., 6.],
       [7., 8.]])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> v
array([1., 1., 1., 1.])
>>> v[:,newaxis]
array([[1.],
       [1.],
       [1.],
       [1.]])
>>> v[:,newaxis].shape
(4, 1)
>>> b.shape
(4, 2)
>>> v.column_stack(v)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'numpy.ndarray' object has no attribute 'column_stack'
>>> np.column_stack((v,v))
array([[1., 1.],
       [1., 1.],
       [1., 1.],
       [1., 1.]])
>>> a = np.arange(4) * 10
>>> a
array([ 0, 10, 20, 30])
>>> np.column_stack((a, a, a))
array([[ 0,  0,  0],
       [10, 10, 10],
       [20, 20, 20],
       [30, 30, 30]])
>>> a np.column_stack((a, a, a))
  File "<stdin>", line 1
    a np.column_stack((a, a, a))
      ^
SyntaxError: invalid syntax
>>> a = np.column_stack((a, a, a))
>>> a
array([[ 0,  0,  0],
       [10, 10, 10],
       [20, 20, 20],
       [30, 30, 30]])
>>> b = np.arange(3)
>>> b
array([0, 1, 2])
>>> a * b
array([[ 0,  0,  0],
       [ 0, 10, 20],
       [ 0, 20, 40],
       [ 0, 30, 60]])
>>> b.shape
(3,)
>>> a.shape
(4, 3)
>>> c = np.arange(4)
>>> c
array([0, 1, 2, 3])
>>> a * c
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (4,3) (4,) 
>>> a * b
array([[ 0,  0,  0],
       [ 0, 10, 20],
       [ 0, 20, 40],
       [ 0, 30, 60]])
>>> a.shape
(4, 3)
>>> b.shape
(3,)
>>> a + b
array([[ 0,  1,  2],
       [10, 11, 12],
       [20, 21, 22],
       [30, 31, 32]])
>>> a
array([[ 0,  0,  0],
       [10, 10, 10],
       [20, 20, 20],
       [30, 30, 30]])
>>> a = np.arange(4) + 1
>>> a
array([1, 2, 3, 4])
>>> b = np.arange(7).reshape(4,-1)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: cannot reshape array of size 7 into shape (4,newaxis)
>>> b = np.arange(8).reshape(4,-1)
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> a
array([1, 2, 3, 4])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> np.sum(a)
10
>>> a.sum()
10
>>> a.max()
4
>>> a.min()
1
>>> b.sum()
28
>>> b.sum(axis = 0)
array([12, 16])
>>> b.sum(axis = 1)
array([ 1,  5,  9, 13])
>>> b.max(axis = 1)
array([1, 3, 5, 7])
>>> b.amax(axis = 1)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'numpy.ndarray' object has no attribute 'amax'
>>> b.argmax(axis = 1)
array([1, 1, 1, 1])
>>> b.max()
7
>>> b.max(axis = 1)
array([1, 3, 5, 7])
>>> b.argmax(axis = 1)
array([1, 1, 1, 1])
>>> a
array([1, 2, 3, 4])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> np.maximum(a,b)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (4,) (4,2) 
>>> np.maximum(a[:,newaxis], b)
array([[1, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> a
array([1, 2, 3, 4])
>>> a[:2]
array([1, 2])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> b.shape
(4, 2)
>>> a[:2].shape
(2,)
>>> np.maximum(a[:2], b)
array([[1, 2],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> a[:2] * b
array([[ 0,  2],
       [ 2,  6],
       [ 4, 10],
       [ 6, 14]])
>>> a[:2] + b
array([[1, 3],
       [3, 5],
       [5, 7],
       [7, 9]])
>>> b / a[:2]
array([[0. , 0.5],
       [2. , 1.5],
       [4. , 2.5],
       [6. , 3.5]])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> b.mean()
3.5
>>> b.sum() / b.size()
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: 'int' object is not callable
>>> b.sum() / b.size
3.5
>>> b.mean(axis = 0)
array([3., 4.])
>>> b.mean(axis = 1)
array([0.5, 2.5, 4.5, 6.5])
>>> b.std()
2.29128784747792
>>> b.std(axis = 0)
array([2.23606798, 2.23606798])
>>> a
array([1, 2, 3, 4])
>>> np.column_stack((a, a))
array([[1, 1],
       [2, 2],
       [3, 3],
       [4, 4]])
>>> np.vstack((a, a))
array([[1, 2, 3, 4],
       [1, 2, 3, 4]])
>>> np.stack((a, a))
array([[1, 2, 3, 4],
       [1, 2, 3, 4]])
>>> np.stack((a, a), axis = 0)
array([[1, 2, 3, 4],
       [1, 2, 3, 4]])
>>> np.stack((a, a), axis = 1)
array([[1, 1],
       [2, 2],
       [3, 3],
       [4, 4]])
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> np.split(b, 2)
[array([[0, 1],
       [2, 3]]), array([[4, 5],
       [6, 7]])]
>>> np.split(b, 2, axis = 1)
[array([[0],
       [2],
       [4],
       [6]]), array([[1],
       [3],
       [5],
       [7]])]
>>> [b1, b2] = np.split(b, 2)
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> b1
array([[0, 1],
       [2, 3]])
>>> b2
array([[4, 5],
       [6, 7]])
>>> b1[0,0] = -1
>>> b1
array([[-1,  1],
       [ 2,  3]])
>>> b2
array([[4, 5],
       [6, 7]])
>>> b
array([[-1,  1],
       [ 2,  3],
       [ 4,  5],
       [ 6,  7]])
>>> from numpy import linalg as la
>>> a
array([1, 2, 3, 4])
>>> la.norm(a)
5.477225575051661
>>> np.sum(a*a)
30
>>> np.sqrt(np.sum(a*a))
5.477225575051661
>>> np.sqrt(a.dot(a))
5.477225575051661
>>> np.sqrt(a @ a)
5.477225575051661
>>> b
array([[-1,  1],
       [ 2,  3],
       [ 4,  5],
       [ 6,  7]])
>>> b[0,0] = 0
>>> b
array([[0, 1],
       [2, 3],
       [4, 5],
       [6, 7]])
>>> la.norm(b)
11.832159566199232
>>> la.norm(b, axis = 0)
array([7.48331477, 9.16515139])
>>> la.norm(b, axis = 1)
array([1.        , 3.60555128, 6.40312424, 9.21954446])
>>> a
array([1, 2, 3, 4])
>>> a[:, newaxis].shape
(4, 1)
>>> aa = a[:, newaxis].shape
>>> aa @ aa.T
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'tuple' object has no attribute 'T'
>>> aa = a[:, newaxis]
>>> aa
array([[1],
       [2],
       [3],
       [4]])
>>> aa @ aa.T
array([[ 1,  2,  3,  4],
       [ 2,  4,  6,  8],
       [ 3,  6,  9, 12],
       [ 4,  8, 12, 16]])
>>> np.outer(a)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "<__array_function__ internals>", line 4, in outer
TypeError: _outer_dispatcher() missing 1 required positional argument: 'b'
>>> np.outer(a, a)
array([[ 1,  2,  3,  4],
       [ 2,  4,  6,  8],
       [ 3,  6,  9, 12],
       [ 4,  8, 12, 16]])
>>> la.matrix_rank(np.outer(a,a))
1
>>> c = np.random.randint(0, 10, (4, 4))
>>> c
array([[7, 5, 7, 4],
       [1, 6, 6, 3],
       [3, 3, 0, 6],
       [3, 9, 7, 0]])
>>> la.matrix_rank(c)
4
>>> la.eig(np.eye(3))
(array([1., 1., 1.]), array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]]))
>>> c
array([[7, 5, 7, 4],
       [1, 6, 6, 3],
       [3, 3, 0, 6],
       [3, 9, 7, 0]])
>>> np.trace(c)
13
>>> qr = la.qr(c)
>>> qr
(array([[-0.84887469,  0.36635641,  0.38083549,  0.01261239],
       [-0.12126781, -0.60936669,  0.33928632, -0.70629395],
       [-0.36380344,  0.04970665, -0.84591815, -0.38678002],
       [-0.36380344, -0.70141604, -0.15579344,  0.59278242]]), array([[-8.24621125, -9.33762156, -9.21635375, -5.94212281],
       [ 0.        , -7.98804253, -6.00161758, -0.06443454],
       [ 0.        ,  0.        ,  3.61101232, -2.53430799],
       [ 0.        ,  0.        ,  0.        , -4.3891124 ]]))
>>> Q = qr[0]
>>> R = qr[1]
>>> Q
array([[-0.84887469,  0.36635641,  0.38083549,  0.01261239],
       [-0.12126781, -0.60936669,  0.33928632, -0.70629395],
       [-0.36380344,  0.04970665, -0.84591815, -0.38678002],
       [-0.36380344, -0.70141604, -0.15579344,  0.59278242]])
>>> R
array([[-8.24621125, -9.33762156, -9.21635375, -5.94212281],
       [ 0.        , -7.98804253, -6.00161758, -0.06443454],
       [ 0.        ,  0.        ,  3.61101232, -2.53430799],
       [ 0.        ,  0.        ,  0.        , -4.3891124 ]])
>>> Q @ R
array([[7.00000000e+00, 5.00000000e+00, 7.00000000e+00, 4.00000000e+00],
       [1.00000000e+00, 6.00000000e+00, 6.00000000e+00, 3.00000000e+00],
       [3.00000000e+00, 3.00000000e+00, 4.99600361e-16, 6.00000000e+00],
       [3.00000000e+00, 9.00000000e+00, 7.00000000e+00, 8.88178420e-16]])
>>> Q[:0] @ Q[:1]
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 1 is different from 4)
>>> Q[:0]
array([], shape=(0, 4), dtype=float64)
>>> Q[0]
array([-0.84887469,  0.36635641,  0.38083549,  0.01261239])
>>> Q[0] @ Q[0]
1.0000000000000002
>>> Q[0] @ Q[1]
1.3877787807814457e-17
>>> np.round(Q[0] @ Q[1])
0.0
>>> la.det(c)
1043.9999999999995
>>> from numpy import linalg as la
>>> numpy.linalg.det(c)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
NameError: name 'numpy' is not defined
>>> np.linalg.det(c)
1043.9999999999995
>>> la.solve
<function solve at 0x7fac3fe51310>
>>> la.inv(c)
array([[ 0.12931034, -0.24137931,  0.03448276,  0.07758621],
       [-0.12356322, -0.08045977,  0.12260536,  0.19252874],
       [ 0.10344828,  0.20689655, -0.17241379, -0.13793103],
       [-0.00287356,  0.16091954,  0.08812261, -0.13505747]])
>>> # Solve the system of equations x0 + 2 * x1 = 1 and 3 * x0 + 5 * x1 = 2
>>> a = np.array([[1, 2], [3, 5]])
>>> b = np.array([1, 2])
>>> x = np.linalg.solve(a, b)
>>> x
array([-1.,  1.])
>>> a @ x
array([1., 2.])
>>>