
     freeware                       xhunga@tiscali.fr
  
     Inner product, Length, Distance.
        
  a) Verify with numeric applications.

     * Properties of Euclidian inner product in R**n.
     * Properties of length   in R**n.
     * Properties of distance in R**n.
     * u.v = 1/4 ||u+v||**2  -  1/4 ||u-v||**2. 
     * Cauchy-Schwarz inequality in R**n.
     * If u.v =0 :  ||u+v||**2 = ||u||**2 + ||v||**2.

  b) The coefficients, integers or fractions, are randomly
     selected by the computer.

  c) The size of the vectors, are randomly selected by the 
     computer, but you can selecte  the size  if you want.
   
      I have used gcc.2.9 (Dev-C++ 4), 
                  gcc.3.2 (v2DJGPP Rhide).

        The list of all  the programs : aaprogr.txt

        Save your work in a text file : aaasave.txt
        ShortName  For  The  Matrices : aashtnm.txt

