You are given that the set {v_{1}, v_{2}, ..., v_{n}} is linearly independent, which means that the equation
c_{1}v_{1} + c_{2}v_{2} + ... + c_{n}v_{n} = 0 has only the trivial solution.
Now look at the equation a_{1}(v_{1}  b) + a_{2}(v_{2}  b) + ... + a_{n}(v_{n}  b) = 0, where b != 0, and b != v_{i}, and show that this equation has only the trivial solution.
