python09():
  Exercises for linear algebra.

  x = 
[1 2 3]
  type(x) =  <class 'numpy.ndarray'>
  x.shape =  (3,)
  len(x) =  3

  x = 
[1 2 3]
  x.T = 
[1 2 3]
   np.transpose(x) = 
[1 2 3]

  r = 
[[1 2 3]]
  type(r) =  <class 'numpy.ndarray'>
  r.shape =  (1, 3)
  len(r) =  1
  r.T = 
[[1]
 [2]
 [3]]

  c = 
[[1]
 [2]
 [3]]
  type(c) =  <class 'numpy.ndarray'>
  c.shape =  (3, 1)
  len(c) =  3
  c.T = 
[[1 2 3]]

  2-norm and max-norms of vectors:

2Norm of [0.58059087 0.55830806]  is  0.8054772771622787
MNorm of [0.58059087 0.55830806]  is  0.5805908708937461
2Norm of [3 4]  is  5.0
MNorm of [3 4]  is  4.0
2Norm of [1. 1.]  is  1.4142135623730951
MNorm of [1. 1.]  is  1.0

  Magnitude and direction of a vector:

  v:
[2 3]
  ||v|| = np.linalg.norm(v) =  3.605551275463989
  vhat = v / ||v||:
[0.5547002  0.83205029]
  v2 = ||v|| * vhat:
[2. 3.]
  w = 10 * v
[20 30]
  ||w|| = np.linalg.norm(w) =  36.05551275463989
  what = w / ||w||:
[0.5547002  0.83205029]

  dot product of w and x: np.dot(w,x)

  w = np.random.rand ( 2 )
[0.94984462 0.43526836]
  x = np.random.rand ( 2 )
[0.23238047 0.78150679]
  np.dot(w,x) =  0.5608905249145545

  Cosine of angle between w and x:  w dot x / ||w|| / ||x||

  ||w|| =  1.0448269498399188
  ||x|| =  0.8153241999651156
  cos(w,x) =  0.658420591366336
  angle =  0.8520779601800194 radians,  48.82047093443133  degrees

  norm(x) = sqrt(x dot x )

  norm ( x ) =  0.8153241999651156
  sqrt(dot(x,x)) =  0.8153241999651156

  projection of v onto u

  u:
[3 4]
  uhat:
[0.6 0.8]
  v:
[5 1]
  beta = np.dot ( v, uhat ) =  3.8
  v1:
[2.28 3.04]
  v2:
[ 2.72 -2.04]
  angle (v1,uhat) =  0.0  =  0.0 degrees
  angle (v2,uhat) =  1.5707963267948966  =  90.0 degrees

  Matrix A:
[[11 12 13 14]
 [21 22 23 24]
 [31 32 33 34]]
  type(A) =  <class 'numpy.ndarray'>
  A.shape =  (3, 4)
  len(A) =  3
  A.T:
[[11 21 31]
 [12 22 32]
 [13 23 33]
 [14 24 34]]

  Matrix norms:

  np.linalg.norm(A) =  83.00602387778854
  np.linalg.norm(A,1) =  72.0
  np.linalg.norm(A,2) =  82.99552942749328
  np.linalg.norm(A,np.inf) =  130.0
  np.linalg.norm(A,'fro') =  83.00602387778854

  Matrix-vector multiplication: A*x=y

A
[[1 2 3]
 [4 5 6]
 [7 8 0]]
x
[1 2 3]
y = np.matmul(A,x) = A*x:
[14 32 23]

  Given A, y, solve A*x=y for x

A
[[1 2 3]
 [4 5 6]
 [7 8 0]]
y
[14 32 23]
x = np.linalg.solve(A,y)
[1. 2. 3.]
  Verify by computing A*x:
[14. 32. 23.]

  Estimate matrix norm ||A|| = max ||Ax|| / ||x||

  A_norm =  13.201458618948678
  A_norm_estimate =  13.193701316232195