%
% Written by:
% -- 
% John L. Weatherwax                2009-04-21
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

F = [ -1, 1, 0; 1, -2, 2; 0, 2, -4 ]; 

% Part (a):
%
ustar = [1;0];

G = [ 1, 0; 1, 1; 0, 1 ]; 

% can we find value of x_1, x_2, and x_3 such that we get F x = -G * ustar ... no 
%
rref( [ F, -G * ustar ] )


% Part (b):
%
ustar = [0;1];

G = [ 1, 0; 1, 1; 0, -2 ]; % this is my problem for part (b) ... make the second column of G in the range of the column space, i.e. this column is = sum(F,1)

% can we find value of x_1, x_2, and x_3 such that we get F x = -G * ustar ... no
%
rref( [ F, -G * ustar ] )


% Part (c):
%
F(3,3) = 0; 
ustar = [1;1];

G = [ 1, 0; 1, 1; 0, 1 ]; 

% can we find value of x_1, x_2, and x_3 such that we get -G * u1 ... yes an exact solution exists
%
F \ (-G * ustar)