Prove that one of the functions is zero. Let the linear functions $l_1, l_2, \ldots , l_k$ is given on a linear space $V$ over an infinite field, and $l_1 (x) \cdot l_2 (x) \cdot \ldots  \cdot l_k (x) = 0$ $\forall $x$ \in V$. Prove that then some of these functions are zero.
 A: This question has a background. My teacher tells me the following problem is a problem in graduate entrance exam in Chinese Tsinghua university.
Let $V_1 \subsetneq V,V_2 \subsetneq V,...V_s \subsetneq V$ be $s$ proper subspace of $V$,prove that there existes a $\alpha \in V$ such that:
$\alpha \notin V_1,V_2,...,V_s$ $(1)$.
You can use induction method. First consider $s=2$, there exists $\beta \notin V_1$.We only need to consider whether $\beta \in V_2$. If $\beta \notin V_2$,then the problem is solved. If $\beta \in V_2$, we consider $\gamma \notin V_2$,if $\gamma \in V_1$, we have:
$\beta \notin V_1$
$\beta \in V_2$
$\gamma \in V_1$
$\gamma \notin V_2$
then $\beta+\gamma$ is required $\alpha$.
Then we can suppose $(1)$ holds for $s-1$. There is an $\alpha$ such that $\alpha \notin V_1,V_2,...,V_{s-1}$.We now only need to consider the case where $\alpha \in V_s$.There is a $\beta \notin V_s$. Consider $s$ vectors $\beta ,\alpha+\beta,...,(s-1)\alpha+\beta $. Obviously these $s$ vectors is not in $V_s$, since $\alpha \in V_s$ but $\beta \notin V_s$.You can prove that there must be a vector in $\beta,\alpha+\beta,...,(s-1)\alpha+\beta $ that does not lie in any one of $V_1,V_2,...,V_{s-1}$ (If every one of these $s$ vectors lie in one of $V_1,V_2,...,V_{s-1}$, how many vectors? How many subspaces? Pigeon hole is followed).
The relation of your question of $(1)$ and your question is obvious.If none of these linear functions vanishing, we consider $V_i = ker(l_i)$ and $V_i$ is a $\textbf{proper}$ subspace of $V$.
At last we consider the question, what if $V$ is free module over a commutative ring with $1$?
