$v_1, v_2, ..., v_n$ are linearly independent
$c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ has a unique of solutions
There does not exists an $i$ such that $v_i$ is a linear combination of $v_1, ..., v_{i-1}, v_{i+1}, ..., v_n$.
Let $A = [v_1 v_2 ... v_n]$
$Ax = 0$ has a unique solution.
$Ax = b$ has at most one solution.
If $A$ is a square matrix, $Ax = b$ has a unique solution.
If $A$ is a square matrix, $det(A) \not= 0$
0 is not an eigenvalue of $A$.