This is a new topic and I've included the only two bits of information we have on this topic below the question. If the answers are related to either of the definition/theorems that I included, could you specify how they apply?

Let V the vector space of polynomials of degree S 100 with the usual addition and scalar multiplication operations. For each of the following subsets W of V, verify either that W is a subspace of V or explain why W is not a subspace of V. (a) when W is the subset of polynomials of degree 5 50. (b) when W is the subset of polynomials in V vanishing at the first 5 positive integers (i.e., the set of polynomials p in V such that 11(1) = 12(2) = = 11(5) = 0)- (c) when W is the subset of polynomials p in V satisfying p(x + 1) = p(x) + l. A subspace of a vector space V is a subset H of V that has three properties: a. The zero vector of V is in H .2 b. H is closed under vector addition. That is, for each u and v in H, the sum 11 + v is in H . c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H. If V1, . . ., Vp are in a vector space V, then Span {V1, ..., Vp} is a subspace of V