Let H be a hyperplane in a linear space X. Then H is parallel to unique subspace

Question:

Let H be a hyperplane in a linear space X. Then H is parallel to unique subspace V such that 1. x0 ∈ V ⇔ H = V
2. V ⊂ X
3. X = lin{V, x1} for every x1 ∉ V
4. for every x ∈ X and x1 ∉ V, there exists a unique α ∈ R such that x = ax1 + v for some v ∈ V