If You acquire the notation:
(x^1, x^2, x^3)
for a generic element of Your space than the condition
x^2 + x^3 = 0
means that the generic element of given subspace is of the form
Observe that A*(1,0,0) = (1,1,1) which is not an element of our subspace...
So, explain the notation or give us some clue