Answer
See answer below
Work Step by Step
Assume that $\{v_1,v_2\}$ is linearly independent (1)
and $v_3 \notin$ span $\{v_1,v_2\}$ .
Let $c_1,c_2,...c_n$ be scalars
$\rightarrow c_1v_1+c_2v_2+...+c_kv_k=0$
Suppose $c_3 \ne 0 \rightarrow v_3=-\frac{c_1}{c_3}v_1-\frac{c_2}{c_3}v_2$ so $c_3$ will $in$ span $\{v_1,v_2\}$
Hence, $c_3=0 \rightarrow c_1v_1+c_2v_2+c_3v_3=c_1v_1+c_2v_2+0=c_1v_1+c_2v_2$
$\rightarrow c_1v_1+c_2v_2=0$ (2)
From (1) and (2) $\rightarrow c_1=c_2=0$
$\rightarrow c_1=c_2=c_3=0$
Hence, the set $\{v_1,v_2,v_3\}$ is linearly independent.
