function Q=mygs4(V)
% Sample function for typical in-class fourth-try to implement
% Graham Schmidt Orthogonalization for general set of col vectors.
% Now using matrix language for the projections as well!
%
% 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)
    w=w-Q(:,1:i-1)*Q(:,1:i-1)'*w % TAKE TIME TO UNDERSTAND THIS!
    Q(:,i)=w/sqrt(w'*w)
end
check1=Q'*Q                % orthogonality
check2=rref([Q,V])         % same span (no new direx added)?