Solving system of linear equations over GF(2)

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You should use the kernel of a linear map (or a matrix, in which case you should specify whether the action is left or right). Both are implemented in Sage.

Here is an example:

sage: A = random_matrix(ZZ,3,4)
sage: A
[  1  -3  -2   0]
[  2   1   1  -1]
[-31   1 -12   1]

sage: b = vector((1,1,1))
sage: a = A.solve_right(b)
sage: a
(26/29, 43/29, -66/29, 0)
sage: A*a == b
True

So, in this case there is a solution to the equation you are trying to solve (if there is no solution, Sage will raise a ValueError: matrix equation has no solutions, so you have to use a try: except: statement to deal with that).

Now, the set of solutions is the set of a+c where c belongs to the kernel of A. You can get this kernel as follows:

sage: K = A.right_kernel()
sage: K
Free module of degree 4 and rank 1 over Integer Ring
Echelon basis matrix:
[ 37  69 -85  58 ]

And check the previous statement:

sage: k = K.basis()[0]
sage: k
(37, 69, -85, 58)
sage: A*k
(0, 0, 0)
sage: A * (a+k) == b
True
sage: A * (a+42*k) == b
True