Consider the following problem:
Let $v_1,v_2,v_3,v_4,v_5\in V$, where $V$ is a vector space over ${\mathbb R}$ and $v_i\neq 0$ for $i = 1,2,3,4,5$. If the following statement is true:
$\sum_{i=1}^{5}a_iv_i\neq0$ whenever $a_i\neq 0 $ for all $i$.
What is the possible least dimension of $V$?
All I have tried so far is considering the contrapositive of the statement. But I have no idea how to go on. Furthermore, can this problem be generalized to the $v_i(i=1,2,\cdots,n)$ case?