In Problems 1–14, a subset W of some n-space Rⁿ is defined by means of a given condition imposed on the typical vector (x₁, x₂, ...., x_n). Apply Theorem 1 to determine whether or not W is a subspace of Rⁿ.

W is the set of all vectors in R³ such that x₃ = 0.

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THEOREM 1 Conditions for a Subspace The nonempty subset W of the vector space V is a subspace of V if and only if it satisfies the following two conditions: (i) If u and v are vectors in W, then u + v is also in W. (ii) If u is in W and c is a scalar, then the vector cu is also in W.