The set, together with the scalar multiplication and vector addition operations, has to satisfy the axioms of a vector space. Now, in your case, you know already (I assume?) that R^2 is a vector space with "the standard operations". When, therefore, you're trying to determine if a set is a vector subspace, your job is a little easier. All you have to do is determine three things:
1. The set is nonempty.
2. The set is closed under scalar multiplication.
3. The set is closed under vector addition.
If those three conditions are satisfied, you've got yourself a subspace. Do those three conditions hold for your set?