Is $\mathbb{P}_n^{-1}[x]$ a vector space? 
Let $\mathbb{P}_n^{-1}[x]=\{a_0+a_1x+a_2x^2+\cdots+a_nx^n:a_0-a_1+\cdots+(-1)^na_n=0\}$.
Is  $\mathbb{P}_n^{-1}[x]$ a vector space?

I took different values of $a_i$ for different polynomials such that they satisfy the condition $a_0-a_1+\cdots+(-1)^na_n=0$, and added them, I think that it does satisfy the condition for vector spaces. Am I correct?

If $P(x)$ is a polynomial with $\deg P(X)\leqslant n$, then$$P(x)\in\Bbb P_n^{-1}[x]\iff P(-1)=0.$$So,

*

*$\Bbb P_n^{-1}[x]$ is not empty, since the null polynomial belongs to it.

*If $P(x),Q(x)\in\Bbb P_n^{-1}[x]$, $(P+Q)(-1)=P(-1)+Q(-1)=0$.

*If $P(x)\in\Bbb P_n^{-1}[x]$ and $\lambda$ is a scalar $(\lambda P)(-1)=\lambda P(-1)=0$.

So, yes, $\Bbb P_n^{-1}[x]$ is a vector space, since it is a subspace of the space of all polynomials $P(x)$ such that $\deg P(x)\leqslant n$.
