I hope this helps:
sage: A = matrix(GF(2), [
[1, 1, 1, 0],
[0, 1, 1, 0],
[1, 0, 0, 1],
[0, 1, 0, 1]])
sage: b = vector(GF(2), [0, 1, 1, 0])
sage: A \ b
(1, 0, 1, 0)
The "x" is indeed in GF(2):
sage: x = A \ b
sage: x.parent()
Vector space of dimension 4 over Finite Field of size 2
Harald
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