Let $V$ be a vector space over a field $F$ and $u_{1},...,u_{n}\in V$ are linearly independent. Prove that for any $v_{1},...,v_{n}\in V$, $u_{1}+\alpha v_{1},...,u_{n}+\alpha v_{n}$ are linearly independent for all but finitely many values of $\alpha\in F$.
I know that if $v_{1},...,v_{n}\in V$ are linearly dependent and there exist $\beta\in F$ such that $u_{1}+\beta v_{1},...,u_{n}+\beta v_{n}$ are linearly dependent, then there are infintely many values to make that set linearly dependent. But I have no idea if $v_{1},...,v_{n}\in V$ are linearly independent, Can someone give me some advise? Thanks!