# coding: utf-8

# # Computing the Nullspace

# In[16]:

import numpy as np
import numpy.linalg as la


# In[17]:

n = 5
np.random.seed(25)
A = np.random.randn(n, n)

# Decrease the rank
A[4] = A[0] + 5 * A[2]
A[1] = 3 * A[0] -2 * A[3]


# In[18]:

from m_echelon import m_echelon


# In[29]:

M, U = m_echelon(A.T)


# In[30]:

la.norm(
    M.dot(A.T) - U)


# In[31]:

U.round(3)


# Now define `NUT` as vectors spanning the nullspace of $N(U^T)$.

# In[32]:

NUT = np.eye(n)[:, 3:]
NUT


# Check that it's actually a nullspace:

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U.T.dot(NUT)


# Now define `NA` as some vectors spanning $N(A)$:

# In[34]:

NA = M.T.dot(NUT)


# And check:

# In[35]:

A.dot(NA)


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