%% Matlab code for in class examples, 6.3:

%% The orthogonal decomposition theorem:
% Let u1, u2 be given as the span of W (they are orthog), and let y be as
% given.  Decompose y as the sum of two vectors- one is the orthog proj of
% y into W, and the other is in W^perp

% First, define the three vectors:
u1=[3;0;1];  u2=[0;1;0];  y=[0;3;10];

% Compute the projection into the span of u1, u2- This is y_hat (or yhat)
yhat=( (y'*u1)/(u1'*u1) )*u1 + ( (y'*u2)/(u2'*u2) )*u2;

% z is computed as y-y_hat:
z=y-yhat;

% We can verify that z is in W^perp by seeing that z is orthog to both u1
% and u2:

z'*u1;

z'*u2;

%% The distance between y and subspace W is the length of z:
norm(z);

% or equivalently,
sqrt(sum(z'*z))

%% The projection of y into W using a matrix U

U=[u1 u2];

%normalize the columns of U
d=sqrt(sum(U.*U))  %This is a row vector whose elements are the norm of 
                   % each column.
d1=repmat(d,3,1)

U=U./d1;

%The projection is:
U*U'*y