# Does multiplying a nonzero scalar to a vector in a linearly independent set preserve linear independence?

Let $\{v_1,v_2,\ldots,v_n\}$ be a linearly independent subset of a vector space $V$ over a field $F$. If $0\ne c\in F$, is $\{cv_1,v_2,\ldots,v_n\}$ linearly independent over $F$?

Yes. $$a_1(cv_1) + a_2v_2 + ... a_nv_n = 0 \implies a_1c=0; a_2=0; ... a_n =0$$ and since $c \neq 0$, the first equality on the right side of the implication boils down to $a_1=0$.
Yes, since any linear dependence between $v_1,\ldots,v_n$ can be turned into a linear dependence between $cv_1,\ldots,v_n$ by dividing the first coefficient by $c$.