% Exploration whether MATLAB can numerically tell 
% the linear dependence of set of the functions 
% {1,cos(x),cos(2x),xos^2(x)}.
% Using 31 datapoints, i.e work w/ a 4  x 31 matrix
% (projection of function space into 31 dim space).
% Some graphical displays....
%
% Primarily for demo purposes to start a discussion.
%
% All rights reserved. Matthias Kawski. 
%                      http://math.la.asu.edu/~kawski
% Original Fall 1996 (?).
% Comments added August 2000.

echo on

xx=[0:0.1:3];
yy=ones(size(xx));
cc=cos(xx);
c2c=cos(2*xx);
csq=cc.*cc;
plot(xx,cc,xx,c2c,xx,csq,xx,yy)
axis([0 3.2 -1.1 1.1])
pause

A(1,:)=yy;
A(2,:)=cc;
A(3,:)=csq;
A(4,:)=c2c

plot(A)
axis([1.8 4.2 -1.1 1.1])
pause

plot(A')
axis([0 32 -1.1 1.1])
pause

rank(A)
pause

rref(A)
pause

help rank
pause

rank(A,1)
rank(A,10)
rank(A,0.1)
rank(A,0.01)
rank(A,0.0001)
pause
rank(A,0.0000001)
rank(A,0.00000000001)
rank(A,0.00000000000001)
rank(A,0.0000000000000001)

echo off