Let and
be two vectors in
. Prove that they are linearly independent if and only if
.
Proof. Assume . Then
implies
Since we know at least one of
and at least one of
are nonzero. Without loss of generality, assume
. Then,
Hence, and
are independent.
Conversely, assume and
are independent. Then the system of equations
has only the trivial solution. We know at least one of and
are non-zero (otherwise one of the vectors is the zero vector, contrary to our hypothesis that the two vectors are independent). Without loss of generality, assume
. As before, we have
But, if , then
can take any value, contradicting that the system has only the trivial solution. Thus,