Proving that the sum $\sum_{x \in \mathbb{F}_2^n} (-1)^{w^Tx}$ is either $2^n$ or zero

Question

Please help me prove that $$\sum_{x \in \mathbb{F}_2^n} (-1)^{w^T \cdot x} = 2^n \delta(w)$$ for any vector $w \in \mathbb{F}_{2}^n$ where $\delta(w) = 1$ for $w = 0$ and $\delta(w) = 0$ otherwise.

Answer 1

For $w = 0$, it's clear that the sum just counts elements of the vector space and so you have $2^n$

If $w \neq 0$, take the support of $w$ and show that there are exactly the same number of subsets with odd size and with even size. Use these to cancel everything else out.