%% Title for the document:  Homework Section 6.5
% Comments for the section

%% Exercise 24, p. 409
% We can use Latex in the comments (if you have had the calc lab)
% 
% $$\mathbf{v}_2=\mathbf{a}_2-\frac{\mathbf{a}_2\cdot\mathbf{v}}{\|{\mathbf v}\|^2}{\mathbf{v}}$$
% 
clear; %It's a good idea to make each piece self-contained
A=[-10 13 7 -11; 2 1 -5 3; -6 3 13 -3; 16 -16 -2 5; 2 1 -5 -7];

% For the solution, fill in the columns of a matrix V.  I'll get you
% started:

V(:,1)=A(:,1);
V(:,1)=V(:,1)/sqrt(V(:,1)'*V(:,1));
V(:,2)=A(:,2)-(A(:,2)'*V(:,1))*V(:,1);

%This last line will tell Matlab to print V to the screen
V
%% Exercise 25, p. 409:  Use your previous code to create QR
clear
A=[-10 13 7 -11; 2 1 -5 3; -6 3 13 -3; 16 -16 -2 5; 2 1 -5 -7];



%% Exercise 26  The problem uses a "for loop" below.

clear
A=[-10 13 7 -11; 2 1 -5 3; -6 3 13 -3; 16 -16 -2 5; 2 1 -5 -7];

% See if you can figure out what each line does.
[nr,nc]=size(A);
q=A(:,1)/norm(A(:,1));
Q=q;
for j=2:nc
   x=A(:,j);
   v=x-Q*Q'*x;
   v=v/norm(v);
   Q=[Q v];
end
R=Q'*A;
clear ans j nc nr q v x
Q
R