How can we show that $f_1f_2...f_k=0$ iff $\exists j$ st $f_j=0$?

Question

Assume $V$ is an n dimensional vector space. $f_1,...f_k\in V^*,v_1,...,v_k\in V$ Define the symmetric k tensor $f_1f_2...f_k(v_1,..,v_k)=\Sigma_{\delta\in S_k}f_{\delta 1}(v_1)...f_{\delta_k}(v_k)$ How can we show that $f_1f_2...f_k=0$ iff $\exists j$ st $f_j=0$?

Answer 1

HINT: Suppose not. Reduce to the case where the $f_i$ are linearly independent. Now choose $v_j$ so that $f_i(v_j)=\delta_{i,j}$.

P.S. The standard notation would be $\text{Sym}(f_1\otimes \dots \otimes f_k)(v_1,\dots,v_k)$.