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The function $L_a(x)=\langle a,x\rangle=a_1 x_1+\cdots+a_n x_n$ is a linear multivariate function of $(x_1,\ldots,x_n)$. The function $$f(x)+\langle a,x\rangle=f(x_1,\ldots,x_n)+a_1 x_1+\cdots+a_n x_n$$ equals $0$ mod 2 if $f(x)=\langle a,x\rangle$ and $1$ mod 2 otherwise. The sum  \hat{f}(a)=\sum_{(x_1,\ldots,x_n) \in \mathbb{F}_2^n} (-1)^{f(x)+\langle ...