% An example of linear dependence
%
% We start with the matrix 
%
%                  1  4 14
%            a =   9  7 39
%                 12 10 54
%
% To input this matrix in Matlab, the statement is
a=[1 4 14;9 7 39;12 10 54];
%
% Next we use an internal function to reduce the matrix
%
r=rref(a);
disp(r);
%
% Obviously the matrix only spans 2-space, or the last column is linearly
% dependent on the first two columns
%
% In order to motivate later concepts 
%
% To do row reductions, we construct a sequence of elementry matrices
% who are designed to produce an identy matrix.  For example, the first
% matrix should be constructed to subtract 9 times row1 from row2 leaving
% a zero in position (2,1) of the matrix.  This matrix is t1
%
t1=[1 0 0;-9 1 0;-12 0 1];
disp(t1);
a1=t1*a;
disp(a1);
%
% The result of the matrix multiplication is a first column where the first
% element is 1 and the rest of the column is zero.  Next, we want to reduce
% the (2,2) to 1 and the remaining elements in this column to zero.  The matrix
% which does this is t2.
%
t2=[1 0 0;0 1 0;0 0 1];t2(1,2)=4/29;t2(2,2)=-1/29;t2(3,2)=-38/29;
disp(t2);
a2=t2*a1;
%
% The resulting matrix is the same row reduced form as given by the rref
% statement.
%
disp(a2);
%
% Next we ask the question:  What happens if the third column is linearly
% independent.  To do this we modify the original matrix by changing the
% (3,3) element form 54 to 70 and again reducing the matrix
%
b=[1 4 14;9 7 39;12 10 70];
r=rref(b);
disp(r);
%
% Because I only changed the (3,3) element, the elementry matrices remain
% unchanged.
%
b1=t1*b;
disp(b1);
b2=t2*b1;
disp(b2);