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\def\N{N}

$v_1, v_2, ..., v_n$ 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 $A$.