Let be nonzero vectors.
- Prove that if and are parallel, then they are linearly dependent.
- Prove that if and are not parallel, then they are linearly independent.
- Proof. If is parallel to then we know there is some nonzero such that . This implies . Thus, there is a nontrivial solution to the equation ; hence, and are linearly dependent
- Proof. Suppose otherwise, that there are nonzero vectors and that are not parallel but that are linearly dependent. Since and are linearly dependent we know there is a nontrivial solution to the equation . But then, we must have (since if either or were zero we would have or , but neither nor are zero) non zero and so
which implies and are parallel, a contradiction