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