Definition, (Inner Product)  Let    and    be complex vectors in  .  Then their inner product is defined to be

        .  

This inner product has the following properties:

The length of a complex vector in   is given by the formula  

        .  

Proofs.

Proof of  P 1.  Left Distributive  .

Proof of  P 2.  Right Distributive  .

Proof of  P 3.  Left Multiple  .

Proof of  P 4.  Right Multiple  .

Proof of  P 5.  Conjugate Symmetry  .

Proof of  P 6.  Nonnegativity  ,  with equality if and only if  .  

        .

And    iff   for k=1,2,...,n  iff   for k=1,2,...,n  iff  .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell