If
is an orthonormal set in a complex inner product space V, then the set
is also an orthonormal set in V.

Question

If 
  is an orthonormal set in a complex inner product space V, then the set 
  is also an orthonormal set in V.

Quizplus · Accepted Answer

Since the two sets differ only in one entry (the third entry), and the dot product of two vectors is invariant under a scalar multiplication of one of the vectors, it follows that the dot product of any two vectors in the second set is the same as that of the corresponding vectors in the first set (up to a conjugation of the third entry). Therefore, the second set is also orthonormal.

If Is an Orthonormal Set in a Complex Inner