# Finding basis of affine space in python?

## Question:

Suppose I am given an affine space as some conjunction of equalities say:

```
x + y + z = 2 && x - 3z = 4
```

which I represent in python as:

```
[ [1,1,1,2] , [1,0,-3,4] ]
```

I would like to find the basis of this affine set. Does python have any such library?

**Little bit of mathematics**:

Let the affine space be given by the matrix equation Ax = b. Let the k vectors {x_1, x_2, .. x_k } be the basis of the nullspace of A i.e. the space represented by Ax = 0. Let y be any particular solution of Ax = b. Then the basis of the affine space represented by Ax = b is given by the (k+1) vectors {y, y + x_1, y + x_2, .. y + x_k }. If there is no particular solution for Ax = b, then return some error message, as the set represented is empty.

For the above equation the matrix equation is:

Ax = b where

A = [[1, 1 , 1] , [1, 0 , -3]]

x = [x , y , z]^T

b = [2, 4]^T

## Answers:

If you are looking for a numerical (i.e. approximate) solution then you can try this:

```
import numpy as np
import scipy
A = np.array([[1,1,1] , [1,0,-3]])
b = np.array([2, 4])
null_sp = scipy.linalg.null_space(A)
x0 = np.linalg.lstsq(A, b, rcond=None)[0][..., None]
aff_basis = np.c_[np.zeros(A.shape[1])[..., None], x] + x0
print(aff_basis)
```

It gives:

```
[[ 1.69230769 1.10395929]
[ 1.07692308 1.86138762]
[-0.76923077 -0.9653469 ]]
```