Let $V$ be a vector space over $\mathbb{F}$, and suppose that $\vec{v}_{1}, \vec{v}_{2}, \dots, \vec{v}_{m} \in V$ are linearly independent. Prove or disprove: For every choice of non-zero constants $\alpha_{1}, \alpha_{2}, \dots, \alpha_{m} \in \mathbb{F}$, the vectors $\alpha_{1} \vec{v}_{1}, \alpha_{2} \vec{v}_{2}, \dots, \alpha_{m} \vec{v}_{m}$ are also linearly independent.
Update: the original proof I came up with was too verbose. It's not even good to look at it. Turns out that I over-thought about the problem...