consider the subset of P2, W = {p(x) ∈ P2 : p(1)p(3) = 0}. Is W is a subspace of P2? If so, prove it. If not, give a demonstrative counterexample.
I'm having a little trouble proving this. I know I'm meant to use the 3 tests for subspaces, but they all seem to just end up being 0, and I can't make a definitive conclusion.
$W$ is not a subspace.
Consider $p=x^2-1$ and $q=x^2-9$
$p(1)p(3)=0(8)=0$ and $q(1)q(3)=-8(0)=0$ so both are in $W$.
But, $$(p+q)(1)(p+q)(3)=(p(1)+q(1))(p(3)+q(3))=p(1)p(3)+ q(1)q(3) + p(1)q(3) + p(3)q(1)=p(3)q(1)=8(-8)=-64\ne 0$$
Therefore, $W$ is not closed under addition and is not a subspace.