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,