By the definition of a subspace a nonempty subset W of a vector space is called subspace of V, if W is a vector space under the operations of addition and scalar multiplication defined in V.

From the above definition it is clear that a subspace must also be a vector space itself and thus must satisfy the axioms of a vector space.

However, the additive identity for \(\displaystyle{R}^{{2}}\) that is,(0,0) is not in the subset W .

Hence, all the vector space axiom are not satisfied by W, so it is not a vector space, and hence not a subspace.

Therefore, W is not a subspace of \(\displaystyle{R}^{{2}}\).

From the above definition it is clear that a subspace must also be a vector space itself and thus must satisfy the axioms of a vector space.

However, the additive identity for \(\displaystyle{R}^{{2}}\) that is,(0,0) is not in the subset W .

Hence, all the vector space axiom are not satisfied by W, so it is not a vector space, and hence not a subspace.

Therefore, W is not a subspace of \(\displaystyle{R}^{{2}}\).