TITLE: Which logical interpretation is correct?
QUESTION [0 upvotes]: The question is: 
Suppose $v_1,..., v_m$ is linearly independent in $V$ and $w \in V$. Show that
$v_1,..., v_m ,w$ is linearly independent if and only if $ w \not\in $ span($v_1,...,v_m$).
Should this be logically interpreted as $v_1,..., v_m$ L.I. $\land $ $v_1,..., v_m ,w$ L.I. $\land$ $w \in V$ $\iff$ $v_1,..., v_m$ L.I.  $\land$ $w \in V$ $\land$ $ w \not\in $ span($v_1,...,v_m$)
or 
$v_1,..., v_m$ L.I. $\land $ $v_1,..., v_m ,w$ L.I. $\land$ $w \in V$ $\iff$ $ w \not\in $ span($v_1,...,v_m$)?

REPLY [2 votes]: It means\begin{multline}\{v_1,\ldots,v_n\}\text{ linearly independent}\implies\\\implies\bigl(\{v_1,\ldots,v_n,w\}\text{ linearly independent}\iff w\notin\operatorname{span}\bigl(\{v_1,\ldots,v_n\}\bigr)\bigr).\end{multline}
{"set_name": "stack_exchange", "score": 0, "question_id": 3256540}