Multilinear polynomials over vectors that span the space

Question

On this Wikipedia article it says "if a multilinear polynomial is zero on a set of vectors that span the space, it will be zero everywhere." Is it referring to the set of vectors that span the domain space, in that any input can be represented as a linear combination of these vectors? If so, this doesn't make sense to me. For example, $f(x,y)=xy$ is a multilinear polynomial, and the vectors $[0,1]$ and $[1,0]$ span the domain space. Since $f(0,1)=f(1,0)=0$ the article claims that $f=0$ but of course $f$ is not the $0$ polynomial.