# Which of the following subsets of P2 are subspaces of P2?

{p in P2: p(0) > p(1)} {p in P2: p(3) = p(4)} {p in P2: p'(3) = 4p(7)}

I understand that generally we need to check to make sure that these are closed under addition and scalar multiplication, and that it is not an empty set. I am having a very difficult time understanding exactly how to show this though, especially because of the inequality in the first question and the derivative in the third question.

The zero polynomial isn't in the first subset so it isn't a subspace of $P_2$. The third subset isn't empty since it contains the zero polynomial. Now let $p$ and $q$ in this subset and $\lambda\in\Bbb R$ then
$$(p+\lambda q)'(3)=p'(3)+\lambda q'(3)=4p(7)+4\lambda q(7)=4(p+\lambda q)(7)$$ hence $p+\lambda q$ is in the third subset. Conclude.