Let $K$ be a field. Consider the vector space $K^n$ over the field $K$. Suppose $(a_1,a_2, ... ,a_n) \in K^n$. What is the dimension of the subspace generated by all the permutations of $(a_1,a_2,...,a_n)$?
I think there are 4 different cases
$a_1=a_2=...=a_n=0$
$a_1=a_2=a_3=...=a_n \ne 0$
$a_1+a_2+...+a_n=0,$ $ a_1 \ne a_2$
$a_1+a_2+...+a_n \ne 0$ , $a_1 \ne a_2$