Let $\mathcal{P}_n[x]$ be the space of polynomials of degree $\leq n$ with real coefficients. I want to find a basis for the subspace of $\mathcal{P}_n[x]$ where the coefficents sum to zero, that is, a space $W$ such that
$$
W = \left\lbrace \sum^n a_k x^k \Bigg| \sum^n a_k = 0 \right\rbrace.
$$
Can one simply use a basis for $\mathcal{P}_n[x]$, for example $\lbrace 1, x, \dots, x^n \rbrace$? My reasoning is that if we have a $u \in W$, then we can write $u$ as a linear combination of elements from $\lbrace 1, x, \dots, x^n \rbrace$ by using the cofficients in $u$.
Best Answer
Hint Notice that for $$p(x) := a_n x^n + \cdots + a_1 x + a_0 \in \mathcal{P}_n[x]$$ the sum $$\sum_{k = 0}^n a_k$$ is just $p(1)$, so $W$ is exactly the kernel of the (linear) map $$e_1 : \mathcal P_n[x] \to \Bbb R, \qquad p \mapsto p(1) .$$ Now, since $e_1$ evaluates any $p \in \mathcal P_n[x]$ at $x = 1$, it is convenient to rewrite $p(x)$ as a linear combination of powers of $$p(x) = \sum_{k = 0}^n b_k (x - 1)^k .$$ What is $W$ in terms of the coefficients $b_k$?