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 solve
vectors $\overline{v},\overline{w}$ are linear independent if
$$ \exists(c_1,c_2)\in \mathbb{R} : c_1\overline{v} + c_2\overline{w} = \overline{0} \implies c_1=0,c_2=0 $$
Now it follows from this that they are not parallel when this condition is satisfied.
However, I'm having trouble connecting the fact that these vectors cannot be parallel when they are linearly independent. This is intuitive to me at some level by the definition.
One way would be to find a connection with cross product and the fact that when
$$ \overline{v} \times \overline{w} = 0 \implies \text{ parallel} $$
then since I wanted to show that they are not parallel use negation
$$ \overline{v} \times \overline{w} \neq 0 \implies \text{ not parallel } $$
But it's problematic since it limits me to $\mathbb{R}^3$ vector space?
Better option is possibly to try to find
$$ \forall(a,b)\in \mathbb{R} : a \overline{v} – b \overline{w} \neq \overline{0} $$
which implies they cannot be parallel since by scaling them with arbitrary $(a,b)$ they cannot be the same.
Best Answer
By the contrapositive, if they are parallel, then there must exist a scalar $\alpha$ such that $\bar{v} - \alpha \bar{w} = 0$.