Problem
Show that if vectors $(\overline{v},\overline{w}) \in V$ are linearly independent and neither of them is zero vector then they are not parallel
Attempt to show by contrapositive
Vectors $(\vec{v},\vec{w})$ are parallel when
$$ \exists(a,b) \in \mathbb{R} \setminus \{0\} : a \vec{v}= b\vec{w} $$
$$ \iff a \vec{v} - b \vec{w} = \vec{0} $$
By definition vectors $(\vec{v},\vec{w})$ are linearly dependent if
$$ ( \exists{c_1,c_2} \in \mathbb{R} : c_1 \vec{v} + c_2 \vec{w} = \vec{0} ) \wedge (c_1 \vee c_2 \neq 0) $$
Then let $c_1 = a, c_2 = -b$ then we have
$$ a\vec{v}-b\vec{w} \iff c_1 \vec{v} + c_2 \vec{w} = \vec{0} $$
Do I have any flaws in this?