Final answer:
Set S, which consists of polynomials satisfying p(1) > p(0), is not a subspace of the vector space P5 because it does not contain the zero vector, is not closed under addition nor under scalar multiplication.
Step-by-step explanation:
Determine whether the given set S is a subspace of the vector space V. In this scenario, V is P5, the space of all polynomials of degree 5 or less, and S is the subset of P5 consisting of those polynomials satisfying the condition p(1) > p(0). To be a subspace, set S must satisfy three criteria: it must contain the zero vector (in this case, the zero polynomial), it must be closed under vector addition, and it must be closed under scalar multiplication.
Testing for Subspace Criteria
Firstly, the zero polynomial does not satisfy the given condition because p(1) = p(0) = 0, therefore, the first criterion is not met. Secondly, if we consider two polynomials p and q in S, such that p(1) > p(0) and q(1) > q(0), we cannot guarantee that their sum p+q will also satisfy (p+q)(1) > (p+q)(0). Lastly, scalar multiplication also fails to guarantee the condition because if we multiply a polynomial p by a negative scalar, the inequality would be reversed. Hence, set S is not a subspace of V because it fails to meet all three subspace criteria.