Prove that if vectors are parallel they are dependent and if they are not parallel then they are independent

Let $A, B \in \mathbb{R}^n$ be nonzero vectors.

Proof. If $A$ is parallel to $B$ then we know there is some nonzero $c \in \mathbb{R}$ such that $B = cA$ . This implies $B - cA =O$ . Thus, there is a nontrivial solution to the equation $xA + yB = O$ ; hence, $A$ and $B$ are linearly dependent $. \qquad \blacksquare$
Proof. Suppose otherwise, that there are nonzero vectors $A$ and $B$ that are not parallel but that are linearly dependent. Since $A$ and $B$ are linearly dependent we know there is a nontrivial solution to the equation $xA + yB = O$ . But then, we must have $x,y$ (since if either $x$ or $y$ were zero we would have $xA = O$ or $yB = O$ , but neither $A$ nor $B$ are zero) non zero and so
$xA + yB = O \quad \implies \quad A = \frac{-y}{x} B$

which implies $A$ and $B$ are parallel, a contradiction $. \qquad \blacksquare$