Answer
See below
Work Step by Step
Given $V=R^3\\
v_1=(1,2,3)\\
v_2=(-3,4,5)\\
v_3=(1,-\frac{4}{3},-\frac{5}{3})$
Let $S=span\{v_1,v_2,v_3\}$
We can notice that $0v_1-\frac{1}{3}v_2=0-\frac{1}{3}v_2=-\frac{1}{3}v_2=-\frac{1}{3}(-3,4,5)=(-1,\frac{-4}{3},\frac{5}{3})$
Thus, $S=span\{v_1,v_2,v_3\}=span\{v_1,v_2\}$
Since $v_1,v_2$ are not proportional, $v_1$ and $v_2$ are linearly independent.
Hence, $\{v_1,v_2\}$ is a linearly independent set spans $S$.
