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?

It is a $(n+1)-1=n$ dimensional subspace because it is the kernel of the following linear form described into the basis $(1,x,x^2,\cdots x^n)$ by:
$$\begin{cases}(a_0,a_1,\cdots a_n) \ & \ \to \ & \ a_0-a_1+\cdots+(-1)^na_n\\
P_n[x]  \ & \ \to \ & \ \mathbb{R}\end{cases}$$
