function [Q,R]=mygs5(V)
% Sample function for typical in-class fifth-try to implement
% Graham Schmidt Orthogonalization = general QR-factorization
% Now also returning the triangular matrix of col'ops 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);
R=eye(n);
for i=1:n
    w=V(:,i);
    r=Q(:,1:i-1)'*w;      % coefficients of the projection
    R(1:i-1,i)=r;
    w=w-Q(:,1:i-1)*r;     % subtract the projection
    len=sqrt(w'*w);       % normalization factor
    Q(:,i)=w/len;         % normalize the new i-th basis vector
    R(i,i)=len;           % multiply old R on the LEFT! by diagmatrix
end
% check1=Q'*Q              % orthogonality
% check2=rref([Q,V])       % same span (no new direx added)?
check = V-Q*R