%% Example of how to turn in your work.
% EXERCISE 1:
% In this example, we are going to take an arbitrary matrix A, and produce
% an orthonormal set of vectors that will form the basis for col(A) using
% Gram Schmidt

A=randn(6,3);  %A random matrix A
V=zeros(size(A));

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

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

V(:,3)=A(:,3)-(A(:,3)'*V(:,2))*V(:,2)-(A(:,3)'*V(:,1))*V(:,1);
V(:,3)=(1/sqrt(V(:,3)'*V(:,3)))*V(:,3);

% Check to see if V is orthonormal
V'*V

%% Exercise 26, p. 401
% Find the distance from b to the col(U), where U is given below.

b=ones(8,1); b(5:8)=-b(5:8);
y=b;
% To keep with the usual notation, I will use y for b, and
% yhat for the projection.

U=[-6 -3 6 1
    -1 2 1 -6
    3 6 3 -2
    6 -3 6 -1
    2 -1 2 3
    -3 6 3 2 
    -2 -1 2 -3
    1 2 1 6];

%We see if U has orthogonal columns:
U'*U

%% 26, continued.
%  Next, find the projection of y onto the columnspace of U.  Divide
%  U by 10 and it becomes orthonormal:
U1=U/10;

yhat=U1*U1'*y;  %Projection of y onto the columns of U
z=y-yhat;

Distance=norm(z)

%% Example of embedding  a plot

clear
clc
load clown

whos

image(X);
colormap(map);