Suppose $v_1,\dots,v_m$ is linearly independent in $V$ and $w \in V$. Show that $v_1,\dots,v_m,w$ is linearly independent if and only if

If $w \in \text{span}(v_1,\dots,v_m)$ then

Then $v_1,\dots,v_m,w$ is linearly dependent because

Now suppose $v_1,\dots,v_m,w$ is linearly independent, clearly $w \notin \text{span}(v_1,\dots,v_n)$ since if it was they'd be dependent as seen above.