Let [v1,v2,v3] be a set of nonzero vectors in r^m such that the (transpose of vi)*vj = 0 when i is not equal to j. show that the set is linearly independent.

Let be the matrix whose columns are , and let be the vector whose components are the constants . Now consider the matrix equation



Multiplying both sides by , we have



More explicitly, we're writing



Multiply both sides by and the left hand side can be written as



We're told that whenever , so we're left with



Each of are nonzero, which means their norms are nonzero, which necessarily implies that , and so the vectors must necessarily be linearly independent.