In [1]: 
from __future__ import division
import numpy as np
import matplotlib.pyplot as pt
In [3]: 
n = 64
omega = exp(2*pi*1j*1./n)

opowers = omega**[1,2,3,4,5]

xlim([-1.2, 1.2]); xlabel("Real")
ylim([-1.2, 1.2]); ylabel("Imaginary")
grid()
gca().set_aspect("equal")
plot(opowers.real, opowers.imag, "o")
Out[3]:
[<matplotlib.lines.Line2D at 0x3d13b50>]
In [4]: 
opowers = omega**arange(n)
plot(opowers.imag)
Out[4]:
[<matplotlib.lines.Line2D at 0x42e71d0>]
In [5]: 
plot((opowers**2).imag)
Out[5]:
[<matplotlib.lines.Line2D at 0x4478b50>]
In [6]: 
plot((opowers**5).imag)
Out[6]:
[<matplotlib.lines.Line2D at 0x477da90>]
In [7]: 
dot(opowers, opowers**3)
Out[7]:
(-1.5210055437364645e-14-3.8691272408186705e-14j)
In [8]: 
dot(opowers**17, opowers**3)
Out[8]:
(-5.7620574978045624e-14-3.8857805861880479e-15j)
In [9]: 
dot(opowers**3, opowers**3)
Out[9]:
(-2.042810365310288e-14-3.0975222387041867e-14j)
In [10]: 
dot((opowers**3).conj(), opowers**3)
Out[10]:
(64.000000000000668+0j)
In [11]: 
vdot(opowers**3, opowers**3)
Out[11]:
(64.000000000000668+0j)

Now in matrix form

In [12]: 
mat = np.vstack(
   [opowers**0, opowers**1, opowers**2,
   opowers**3, opowers**4, opowers**5]).T
mat.shape
Out[12]:
(64, 6)
In [13]: 
plot(np.dot(mat, [0, 1, 0,0, 0.3, 0.7]))

/usr/lib/python2.7/dist-packages/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part
  return array(a, dtype, copy=False, order=order)
Out[13]:
[<matplotlib.lines.Line2D at 0x4986090>]
In [14]: 
mat = opowers[:, newaxis] ** arange(n)
mat.shape
Out[14]:
(64, 64)
In [15]: 
weights = zeros(n)
weights[1] = 1
weights[4] = 0.3
weights[5] = 0.7
plot(dot(mat, weights))
Out[15]:
[<matplotlib.lines.Line2D at 0x4bce2d0>]

Inverse of mat?

In [16]: 
dot(mat[:,3].conj(), mat[3])
Out[16]:
(64.000000000000455+9.7574726076743483e-14j)
In []: