%% Quiz 10 Solutions

%% Exercise 24, p. 409
% 
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);
V(:,2)=V(:,2)/sqrt(V(:,2)'*V(:,2));
V(:,3)=A(:,3)-(A(:,3)'*V(:,1))*V(:,1)-(A(:,3)'*V(:,2))*V(:,2);
V(:,3)=V(:,3)/sqrt(V(:,3)'*V(:,3));
V(:,4)=A(:,4)-(A(:,4)'*V(:,1))*V(:,1)-(A(:,4)'*V(:,2))*V(:,2)-(A(:,4)'*V(:,3))*V(:,3);
V(:,4)=V(:,4)/sqrt(V(:,4)'*V(:,4));  

%Check- This should be the 4 x 4 identity matrix
V'*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];

V(:,1)=A(:,1);
V(:,1)=V(:,1)/sqrt(V(:,1)'*V(:,1));
V(:,2)=A(:,2)-(A(:,2)'*V(:,1))*V(:,1);
V(:,2)=V(:,2)/sqrt(V(:,2)'*V(:,2));
V(:,3)=A(:,3)-(A(:,3)'*V(:,1))*V(:,1)-(A(:,3)'*V(:,2))*V(:,2);
V(:,3)=V(:,3)/sqrt(V(:,3)'*V(:,3));
V(:,4)=A(:,4)-(A(:,4)'*V(:,1))*V(:,1)-(A(:,4)'*V(:,2))*V(:,2)-(A(:,4)'*V(:,3))*V(:,3);
V(:,4)=V(:,4)/sqrt(V(:,4)'*V(:,4));  
Q=V;
R=Q'*A;
%Check:
Q*R

%% 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);           %computes the number of rows, cols of A
q=A(:,1)/norm(A(:,1));     % initializes the first column of Q
Q=q;
for j=2:nc                 %Loop
   x=A(:,j);               %  Take the jth column of A
   v=x-Q*Q'*x;             %   Remove the components of x that are in Col(Q)
   v=v/norm(v);            %   Normalize v -  A good idea!
   Q=[Q v];                %   Add that column to the rest.
end                        %End the loop
R=Q'*A;                    % Compute R

clear ans j nc nr q v x
Q
R
%% Hanford Data

load HanfordData1

% We have 9 data points- Index (the indep variable) and Deaths (dependent)
x=Index; y=Deaths;
A=[x(:), ones(9,1)];

% Find the slope and intercept, and label then as m, b

B=inv(A'*A)*A'*y';
m=B(1)
b=B(2)

% Plotting routine (using the m, b found previously)
figure(1)
xx=linspace(min(x),max(x),200);
yy= m*xx + b;
plot(x,y,'b^',xx,yy,'k-');  %See the help file for plot to see what this means

%% Data Set 2:  Knee Girth to Height
% Download the data set Body1.mat from the class website

clear;  %Clear the old variables out
load Body1

% Some plotting commands to see the data first:
figure(2)
plot(KneeGirth,Height,'.')
plot(KneeGirth(menidx),Height(menidx),'r.',KneeGirth(womenidx),Height(womenidx),'b.');
legend('Men','Women')

x=KneeGirth; y=Height;
A=[KneeGirth,ones(size(KneeGirth))];

% Get the slope and intercept, and label them as m, b

B=inv(A'*A)*A'*y;
m=B(1)
b=B(2)


% Once m, b are found, you can run this code:
xx=linspace(min(x),max(x),200);
yy= m*xx + b;
figure(3);  %Opens a new figure for the new plot
plot(x,y,'b.',xx,yy,'k-');