Let vector space $W$ be $\{p(z)\in P_{3}(\mathbb{C}):p(a)=0 \Rightarrow a\in \mathbb{R}\}$ under usual addition and usual scalar multiplication.
How to show that W is not a vector space over $\mathbb{C}$?
It should violate one or more of the 10 axioms of complex vector space, but I am stuck here, I tried and it all seems that 10 axioms holds, how can W not be an complex vector space?
Best Answer
Jose already picked out an answer, but it’d also suffice to observe that the zero polynomial isn’t in $W$.