vectors():
  Some linear algebra exercises with vectors.

  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 [0.08708772 0.63028344]  is  0.6362715502230434
MNorm of [0.08708772 0.63028344]  is  0.630283440495408
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 )
[0.45091203 0.29213392]
  x = np.random.rand ( 2 )
[0.69490582 0.60677313]
  np.dot(w,x) =  0.4906004021174127

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

  ||w|| =  0.5372744951920453
  ||x|| =  0.9225333170099113
  cos(w,x) =  0.9898049366520131
  angle =  0.1429155962005503 radians,  8.188460488887436  degrees

  norm(x) = sqrt(x dot x )

  norm ( x ) =  0.9225333170099113
  sqrt(dot(x,x)) =  0.9225333170099113

  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  radians =  0.0 degrees
  angle (v2,uhat) =  1.5707963267948966  radians =  90.0 degrees