function Q=mygs3(V)
% Sample function for typical in-class third-try to implement
% Graham Schmidt Orthogonalization in one special example.
% Now using for-loop and more matrix language.
%
% All rights reserved.    Original version: Nov 2003
% Matthias Kawski         http://math.asu.edu/~kawski

if nargin<1                  % self-executing --- when no data
    V=5-round(10*rand(7,4))  % are provided, then create and 
end                          % work random example
Q=V
n=size(V,2)
for i=1:n
    w=V(:,i)
    p=0
    for j=1:i-1                   % see mygs4 for parallel
        p=p+(Q(:,j)'*w)*Q(:,j)    % implementation using
    end                           % more matrix language
    w=w-p
    Q(:,i)=w/sqrt(w'*w)
end
check1=Q'*Q                % orthogonality
check2=rref([Q,V])         % same span (no new direx added)?